University Calculus: Early Transcendentals (4th Edition)
4th Edition
ISBN: 9780134995540
Author: Joel R. Hass, Christopher E. Heil, Przemyslaw Bogacki, Maurice D. Weir, George B. Thomas Jr.
Publisher: PEARSON
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Chapter 15.8, Problem 8E
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Calculate the divergence of the field
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Consider the function f(x) = 2x² - 8x + 3 over the interval 0 ≤ x ≤ 9. Complete the following
steps to find the global (absolute) extrema on the interval. Answer exactly. Separate multiple answers
with a comma.
a. Find the derivative of f (x) = 2x² - 8x+3
f'(x)
b. Find any critical point(s) c within the intervl 0 < x < 9. (Enter as reduced fraction as needed)
c. Evaluate the function at the critical point(s). (Enter as reduced fraction as needed. Enter DNE if
none of the critical points are inside the interval)
f(c)
d. Evaluate the function at the endpoints of the interval 0 ≤ x ≤ 9.
f(0)
f(9)
e. Based on the above results, find the global extrema on the interval and where they occur.
The global maximum value is
at a
The global minimum value is
at x
Determine the values and locations of the global (absolute) and local extrema on the graph given.
Assume the domain is a closed interval and the graph represents the entirety of the function.
3
y
-6-5-4-3
2
1
-1
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-3
Separate multiple answers with a comma.
Global maximum: y
Global minimum: y
Local maxima: y
Local minima: y
x
6
at a
at a
at x=
at x=
A ball is thrown into the air and its height (in meters) is given by h (t)
in seconds.
-4.92 + 30t+1, where t is
a. After how long does the ball reach its maximum height? Round to 2 decimal places.
seconds
b. What is the maximum height of the ball? Round to 2 decimal places.
meters
Chapter 15 Solutions
University Calculus: Early Transcendentals (4th Edition)
Ch. 15.1 - Match the vector equations in Exercises 1–8 with...Ch. 15.1 - Match the vector equations in Exercises 1–8 with...Ch. 15.1 - Match the vector equations in Exercises 1–8 with...Ch. 15.1 - Match the vector equations in Exercises 1–8 with...Ch. 15.1 - Match the vector equations in Exercises 1–8 with...Ch. 15.1 - Match the vector equations in Exercises 1–8 with...Ch. 15.1 - Match the vector equations in Exercises 1–8 with...Ch. 15.1 - Prob. 8ECh. 15.1 - Evaluate ∫C (x + y) ds, where C is the...Ch. 15.1 - Evaluate ∫C (x − y + z − 2) ds, where C is the...
Ch. 15.1 - Evaluate ∫C (xy + y + z) ds along the curve r(t) =...Ch. 15.1 - Evaluate Cx2+y2ds along the curve r(t) = (4 cos...Ch. 15.1 - Find the line integral of f(x, y, z) = x + y + z...Ch. 15.1 - Find the line integral of over the curve r(t) =...Ch. 15.1 - Integrate over the path C1 followed by C2 from...Ch. 15.1 - Prob. 16ECh. 15.1 - Integrate f(x, y, z) = (x + y + z)/(x2+ y2+ z2)...Ch. 15.1 - Integrate over the circle r(t) = (a cos t)j + (a...Ch. 15.1 - Evaluate ∫C x ds, where C is
the straight-line...Ch. 15.1 - Evaluate , where C is
the straight-line segment x...Ch. 15.1 - Find the line integral of along the curve r(t) =...Ch. 15.1 - Prob. 22ECh. 15.1 - Prob. 23ECh. 15.1 - Find the line integral of along the curve , 1/2 ≤...Ch. 15.1 - Prob. 25ECh. 15.1 - Prob. 26ECh. 15.1 - Prob. 27ECh. 15.1 - Prob. 28ECh. 15.1 - In Exercises 27–30, integrate f over the given...Ch. 15.1 - In Exercises 27–30, integrate f over the given...Ch. 15.1 - Prob. 31ECh. 15.1 - Prob. 32ECh. 15.1 - Mass of a wire Find the mass of a wire that lies...Ch. 15.1 - Center of mass of a curved wire A wire of density ...Ch. 15.1 - Mass of wire with variable density Find the mass...Ch. 15.1 - Center of mass of wire with variable density Find...Ch. 15.1 - Prob. 37ECh. 15.1 - Prob. 38ECh. 15.1 - Prob. 39ECh. 15.1 - Wire of constant density A wire of constant...Ch. 15.1 - Prob. 41ECh. 15.1 - Prob. 42ECh. 15.2 - Find the gradient fields of the functions in...Ch. 15.2 - Find the gradient fields of the functions in...Ch. 15.2 - Find the gradient fields of the functions in...Ch. 15.2 - Find the gradient fields of the functions in...Ch. 15.2 - Give a formula F = M(x, y)i + N(x, y)j for the...Ch. 15.2 - Give a formula F = M(x, y)i + N(x, y)j for the...Ch. 15.2 - In Exercises 7−12, find the line integrals of F...Ch. 15.2 - In Exercises 7−12, find the line integrals of F...Ch. 15.2 - In Exercises 7−12, find the line integrals of F...Ch. 15.2 - In Exercises 7−12, find the line integrals of F...Ch. 15.2 - Line Integrals of Vector Fields
In Exercises 7−12,...Ch. 15.2 - Line Integrals of Vector Fields
In Exercises 7−12,...Ch. 15.2 - In Exercises 1316, find the line integrals along...Ch. 15.2 - In Exercises 13–16, find the line integrals along...Ch. 15.2 - In Exercises 13–16, find the line integrals along...Ch. 15.2 - In Exercises 13–16, find the line integrals along...Ch. 15.2 - Along the curve , , evaluate each of the following...Ch. 15.2 - Along the curve , , evaluate each of the following...Ch. 15.2 - In Exercises 19–22, find the work done by F over...Ch. 15.2 - In Exercises 19–22, find the work done by F over...Ch. 15.2 - In Exercises 19–22, find the work done by F over...Ch. 15.2 - In Exercises 19–22, find the work done by F over...Ch. 15.2 - Evaluate along the curve from (–1, 1) to (2,...Ch. 15.2 - Evaluate counterclockwise around the triangle...Ch. 15.2 - Evaluate CFTds for the vector field F=x2iyj along...Ch. 15.2 - Evaluate for the vector field counterclockwise...Ch. 15.2 - Work Find the work done by the force F = xyi + (y...Ch. 15.2 - Work Find the work done by the gradient of f(x, y)...Ch. 15.2 - Circulation and flux Find the circulation and flux...Ch. 15.2 - Flux across a circle Find the flux of the...Ch. 15.2 - In Exercises 31–34, find the circulation and flux...Ch. 15.2 - In Exercises 31–34, find the circulation and flux...Ch. 15.2 - In Exercises 31–34, find the circulation and flux...Ch. 15.2 - In Exercises 31–34, find the circulation and flux...Ch. 15.2 - Flow integrals Find the flow of the velocity field...Ch. 15.2 - Flux across a triangle Find the flux of the field...Ch. 15.2 - The flow of a gas with a density of over the...Ch. 15.2 - The flow of a gas with a density of over the...Ch. 15.2 - Find the flow of the velocity field F = y2i + 2xyj...Ch. 15.2 - Find the circulation of the field F = yi + (x +...Ch. 15.2 - Prob. 41ECh. 15.2 - Prob. 42ECh. 15.2 - Prob. 43ECh. 15.2 - Prob. 44ECh. 15.2 - Prob. 45ECh. 15.2 - Prob. 46ECh. 15.2 - Spin field Draw the spin field
(see Figure 15.13)...Ch. 15.2 - Prob. 48ECh. 15.2 - Prob. 49ECh. 15.2 - Prob. 50ECh. 15.2 - Prob. 51ECh. 15.2 - Prob. 52ECh. 15.2 - Prob. 53ECh. 15.2 - Prob. 54ECh. 15.2 - Prob. 55ECh. 15.2 - Prob. 56ECh. 15.2 - Prob. 57ECh. 15.2 - Prob. 58ECh. 15.2 - Prob. 59ECh. 15.2 - Prob. 60ECh. 15.2 - Flow along a curve The field F = xyi + yj − yzk is...Ch. 15.2 - Prob. 62ECh. 15.3 - Which fields in Exercises 1–6 are conservative,...Ch. 15.3 - Which fields in Exercises 1–6 are conservative,...Ch. 15.3 - Which fields in Exercises 1–6 are conservative,...Ch. 15.3 - Which fields in Exercises 1–6 are conservative,...Ch. 15.3 - Which fields in Exercises 1−6 are conservative,...Ch. 15.3 - Which fields in Exercises 1−6 are conservative,...Ch. 15.3 - Finding Potential Functions In Exercises 712, find...Ch. 15.3 -
In Exercises 7–12, find a potential function f...Ch. 15.3 - In Exercises 7–12, find a potential function f for...Ch. 15.3 - In Exercises 7–12, find a potential function f for...Ch. 15.3 - In Exercises 7–12, find a potential function f for...Ch. 15.3 - In Exercises 7–12, find a potential function f for...Ch. 15.3 - In Exercises 13–17, show that the differential...Ch. 15.3 - In Exercises 13–17, show that the differential...Ch. 15.3 - In Exercises 13–17, show that the differential...Ch. 15.3 - In Exercises 13–17, show that the differential...Ch. 15.3 - In Exercises 13–17, show that the differential...Ch. 15.3 - Although they are not defined on all of space R3,...Ch. 15.3 - Prob. 19ECh. 15.3 - Prob. 20ECh. 15.3 - Prob. 21ECh. 15.3 - Prob. 22ECh. 15.3 - Prob. 23ECh. 15.3 - Prob. 24ECh. 15.3 - Prob. 25ECh. 15.3 - Prob. 26ECh. 15.3 - In Exercises 27 and 28, find a potential function...Ch. 15.3 - In Exercises 27 and 28, find a potential function...Ch. 15.3 - Work along different paths Find the work done by F...Ch. 15.3 - Work along different paths Find the work done by F...Ch. 15.3 - Evaluating a work integral two ways Let F =...Ch. 15.3 - Prob. 32ECh. 15.3 - Exact differential form How are the constants a,...Ch. 15.3 - Prob. 34ECh. 15.3 - Prob. 35ECh. 15.3 - Prob. 36ECh. 15.3 - Prob. 37ECh. 15.3 - Prob. 38ECh. 15.4 - In Exercises 1–6, find the k-component of curl(F)...Ch. 15.4 - Prob. 2ECh. 15.4 - Prob. 3ECh. 15.4 - Prob. 4ECh. 15.4 - In Exercises 1–6, find the k-component of curl(F)...Ch. 15.4 - Prob. 6ECh. 15.4 - In Exercises 710, verify the conclusion of Green’s...Ch. 15.4 - In Exercises 7–10, verify the conclusion of...Ch. 15.4 - In Exercises 7–10, verify the conclusion of...Ch. 15.4 - In Exercises 7–10, verify the conclusion of...Ch. 15.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 15.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 15.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 15.4 - Prob. 14ECh. 15.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 15.4 - Prob. 16ECh. 15.4 - Prob. 17ECh. 15.4 - Prob. 18ECh. 15.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 15.4 - In Exercises 11–20, use Green’s Theorem to find...Ch. 15.4 - Find the counterclockwise circulation and outward...Ch. 15.4 - Prob. 22ECh. 15.4 - Prob. 23ECh. 15.4 - Prob. 24ECh. 15.4 - Prob. 25ECh. 15.4 - Prob. 26ECh. 15.4 - Apply Green’s Theorem to evaluate the integrals in...Ch. 15.4 - Prob. 28ECh. 15.4 - Apply Green’s Theorem to evaluate the integrals in...Ch. 15.4 - Apply Green’s Theorem to evaluate the integrals in...Ch. 15.4 - Prob. 31ECh. 15.4 - Prob. 32ECh. 15.4 - Prob. 33ECh. 15.4 - Prob. 34ECh. 15.4 - Prob. 35ECh. 15.4 - Prob. 36ECh. 15.4 - Prob. 37ECh. 15.4 - Prob. 38ECh. 15.4 - Prob. 39ECh. 15.4 - Prob. 40ECh. 15.4 - Prob. 41ECh. 15.4 - Prob. 42ECh. 15.4 - Prob. 43ECh. 15.4 - Prob. 44ECh. 15.4 - Regions with many holes Green’s Theorem holds for...Ch. 15.4 - Prob. 46ECh. 15.4 - Prob. 47ECh. 15.4 - Prob. 48ECh. 15.5 - In Exercises 1–16, find a parametrization of the...Ch. 15.5 - Prob. 2ECh. 15.5 - Prob. 3ECh. 15.5 - Prob. 4ECh. 15.5 - In Exercises 1–16, find a parametrization of the...Ch. 15.5 - Prob. 6ECh. 15.5 - In Exercises 1–16, find a parametrization of the...Ch. 15.5 - Prob. 8ECh. 15.5 - Prob. 9ECh. 15.5 - Prob. 10ECh. 15.5 - In Exercises 1–16, find a parametrization of the...Ch. 15.5 - Prob. 12ECh. 15.5 - In Exercises 1–16, find a parametrization of the...Ch. 15.5 - Prob. 14ECh. 15.5 - Prob. 15ECh. 15.5 - Prob. 16ECh. 15.5 - In Exercises 17–26, use a parametrization to...Ch. 15.5 - Prob. 18ECh. 15.5 - Prob. 19ECh. 15.5 - Prob. 20ECh. 15.5 - Prob. 21ECh. 15.5 - In Exercises 17–26, use a parametrization to...Ch. 15.5 - Prob. 23ECh. 15.5 - In Exercises 17–26, use a parametrization to...Ch. 15.5 - Prob. 25ECh. 15.5 - In Exercises 17–26, use a parametrization to...Ch. 15.5 - Prob. 27ECh. 15.5 - Prob. 28ECh. 15.5 - Prob. 29ECh. 15.5 - Prob. 30ECh. 15.5 - Prob. 31ECh. 15.5 - Prob. 32ECh. 15.5 - Parametrization of an ellipsoid The...Ch. 15.5 - Prob. 34ECh. 15.5 - Prob. 35ECh. 15.5 - Prob. 36ECh. 15.5 - Prob. 37ECh. 15.5 - Prob. 38ECh. 15.5 - Prob. 39ECh. 15.5 - Prob. 40ECh. 15.5 - Prob. 41ECh. 15.5 - Prob. 42ECh. 15.5 - Prob. 43ECh. 15.5 - Find the area of the upper portion of the cylinder...Ch. 15.5 - Prob. 45ECh. 15.5 - Prob. 46ECh. 15.5 - Prob. 47ECh. 15.5 - Prob. 48ECh. 15.5 - Prob. 49ECh. 15.5 - Prob. 50ECh. 15.5 - Prob. 51ECh. 15.5 - Prob. 52ECh. 15.5 - Prob. 53ECh. 15.5 - Prob. 54ECh. 15.5 - Prob. 55ECh. 15.5 - Prob. 56ECh. 15.6 - In Exercises 1–8, integrate the given function...Ch. 15.6 - In Exercises 18, integrate the given function over...Ch. 15.6 - In Exercises 1–8, integrate the given function...Ch. 15.6 - In Exercises 1–8, integrate the given function...Ch. 15.6 - Prob. 5ECh. 15.6 - Prob. 6ECh. 15.6 - Prob. 7ECh. 15.6 - Prob. 8ECh. 15.6 - Prob. 9ECh. 15.6 - Prob. 10ECh. 15.6 - Prob. 11ECh. 15.6 - Prob. 12ECh. 15.6 - Prob. 13ECh. 15.6 - Prob. 14ECh. 15.6 - Integrate G(x, y, z) = z − x over the portion of...Ch. 15.6 - Prob. 16ECh. 15.6 - Prob. 17ECh. 15.6 - Prob. 18ECh. 15.6 - In Exercises 19–28, use a parametrization to find...Ch. 15.6 - Prob. 20ECh. 15.6 - Prob. 21ECh. 15.6 - Prob. 22ECh. 15.6 - Prob. 23ECh. 15.6 - Prob. 24ECh. 15.6 - Prob. 25ECh. 15.6 - Prob. 26ECh. 15.6 - In Exercises 19–28, use a parametrization to find...Ch. 15.6 - Prob. 28ECh. 15.6 - Prob. 29ECh. 15.6 - Prob. 30ECh. 15.6 - Prob. 31ECh. 15.6 - Prob. 32ECh. 15.6 - Prob. 33ECh. 15.6 - Prob. 34ECh. 15.6 - Prob. 35ECh. 15.6 - Prob. 36ECh. 15.6 - Find the flux of the field through the surface...Ch. 15.6 - Prob. 38ECh. 15.6 - Prob. 39ECh. 15.6 - Prob. 40ECh. 15.6 - Prob. 41ECh. 15.6 - Prob. 42ECh. 15.6 - Prob. 43ECh. 15.6 - Prob. 44ECh. 15.6 - Prob. 45ECh. 15.6 - Prob. 46ECh. 15.6 - Prob. 47ECh. 15.6 - Prob. 48ECh. 15.6 - Prob. 49ECh. 15.6 - Prob. 50ECh. 15.7 - Prob. 1ECh. 15.7 - Prob. 2ECh. 15.7 - Prob. 3ECh. 15.7 - Prob. 4ECh. 15.7 - Prob. 5ECh. 15.7 - Prob. 6ECh. 15.7 - In Exercises 7–12, use the surface integral in...Ch. 15.7 - Prob. 8ECh. 15.7 - Prob. 9ECh. 15.7 - Prob. 10ECh. 15.7 - Prob. 11ECh. 15.7 - Prob. 12ECh. 15.7 - Prob. 13ECh. 15.7 - Prob. 14ECh. 15.7 - Prob. 15ECh. 15.7 - Prob. 16ECh. 15.7 - Prob. 17ECh. 15.7 - Prob. 18ECh. 15.7 - In Exercises 19–24, use the surface integral in...Ch. 15.7 - Prob. 20ECh. 15.7 - In Exercises 19–24, use the surface integral in...Ch. 15.7 - Prob. 22ECh. 15.7 - Prob. 23ECh. 15.7 - Prob. 24ECh. 15.7 - Prob. 25ECh. 15.7 - Verify Stokes’ Theorem for the vector field F =...Ch. 15.7 - Zero circulation Use Equation (8) and Stokes’...Ch. 15.7 - Prob. 28ECh. 15.7 - Prob. 29ECh. 15.7 - Prob. 30ECh. 15.7 - Prob. 31ECh. 15.7 - Does Stokes’ Theorem say anything special about...Ch. 15.7 - Let R be a region in the xy-plane that is bounded...Ch. 15.7 - Zero curl, yet the field is not conservative Show...Ch. 15.8 - Prob. 1ECh. 15.8 - Prob. 2ECh. 15.8 - Prob. 3ECh. 15.8 - Prob. 4ECh. 15.8 - Prob. 5ECh. 15.8 - Prob. 6ECh. 15.8 - Prob. 7ECh. 15.8 - Prob. 8ECh. 15.8 - Prob. 9ECh. 15.8 - In Exercises 920, use the Divergence Theorem to...Ch. 15.8 - Prob. 11ECh. 15.8 - Prob. 12ECh. 15.8 - Prob. 13ECh. 15.8 - Prob. 14ECh. 15.8 - Prob. 15ECh. 15.8 - Prob. 16ECh. 15.8 - Prob. 17ECh. 15.8 - Prob. 18ECh. 15.8 - Prob. 19ECh. 15.8 - Prob. 20ECh. 15.8 - Prob. 21ECh. 15.8 - Prob. 22ECh. 15.8 - Prob. 23ECh. 15.8 - Prob. 24ECh. 15.8 - Prob. 25ECh. 15.8 - Prob. 26ECh. 15.8 - Prob. 27ECh. 15.8 - Compute the net outward flux of the vector field F...Ch. 15.8 - Prob. 29ECh. 15.8 - Prob. 30ECh. 15.8 - Prob. 31ECh. 15.8 - Prob. 32ECh. 15.8 - Prob. 33ECh. 15.8 - Prob. 34ECh. 15.8 - Prob. 35ECh. 15.8 - Prob. 36ECh. 15 - Prob. 1GYRCh. 15 - Prob. 2GYRCh. 15 - Prob. 3GYRCh. 15 - Prob. 4GYRCh. 15 - Prob. 5GYRCh. 15 - Prob. 6GYRCh. 15 - What is special about path independent fields?
Ch. 15 - Prob. 8GYRCh. 15 - Prob. 9GYRCh. 15 - Prob. 10GYRCh. 15 - Prob. 11GYRCh. 15 - Prob. 12GYRCh. 15 - What is an oriented surface? What is the surface...Ch. 15 - Prob. 14GYRCh. 15 - Prob. 15GYRCh. 15 - Prob. 16GYRCh. 15 - Prob. 17GYRCh. 15 - Prob. 18GYRCh. 15 - Prob. 1PECh. 15 - The accompanying figure shows three polygonal...Ch. 15 - Prob. 3PECh. 15 - Prob. 4PECh. 15 - Prob. 5PECh. 15 - Prob. 6PECh. 15 - Prob. 7PECh. 15 - Prob. 8PECh. 15 - Prob. 9PECh. 15 - Prob. 10PECh. 15 - Prob. 11PECh. 15 - Prob. 12PECh. 15 - Prob. 13PECh. 15 - Prob. 14PECh. 15 - Prob. 15PECh. 15 - Prob. 16PECh. 15 - Prob. 17PECh. 15 - Prob. 18PECh. 15 - Prob. 19PECh. 15 - Prob. 20PECh. 15 - Prob. 21PECh. 15 - Prob. 22PECh. 15 - Prob. 23PECh. 15 - Prob. 24PECh. 15 - Prob. 25PECh. 15 - Prob. 26PECh. 15 - Prob. 27PECh. 15 - Prob. 28PECh. 15 - Prob. 29PECh. 15 - Prob. 30PECh. 15 - Prob. 31PECh. 15 - Prob. 32PECh. 15 - Prob. 33PECh. 15 - Prob. 34PECh. 15 - Prob. 35PECh. 15 - Prob. 36PECh. 15 - Prob. 37PECh. 15 - Prob. 38PECh. 15 - Prob. 39PECh. 15 - Prob. 40PECh. 15 - Prob. 41PECh. 15 - Prob. 42PECh. 15 - Prob. 43PECh. 15 - Prob. 44PECh. 15 - Prob. 45PECh. 15 - Prob. 46PECh. 15 - Prob. 47PECh. 15 - Moment of inertia of a cube Find the moment of...Ch. 15 - Prob. 49PECh. 15 - Prob. 50PECh. 15 - Prob. 51PECh. 15 - Prob. 52PECh. 15 - Prob. 53PECh. 15 - In Exercises 53–56, find the outward flux of F...Ch. 15 - Prob. 55PECh. 15 - In Exercises 53–56, find the outward flux of F...Ch. 15 - Hemisphere, cylinder, and plane Let S be the...Ch. 15 - Prob. 58PECh. 15 - Prob. 59PECh. 15 - Prob. 60PECh. 15 - Prob. 1AAECh. 15 - Use the Green’s Theorem area formula in Exercises...Ch. 15 - Prob. 3AAECh. 15 - Use the Green’s Theorem area formula in Exercises...Ch. 15 - Prob. 5AAECh. 15 - Prob. 6AAECh. 15 - Prob. 7AAECh. 15 - Find the mass of a helicoids
r(r, ) = (r cos )i +...Ch. 15 - Prob. 9AAECh. 15 - Prob. 10AAECh. 15 - Prob. 11AAECh. 15 - Prob. 12AAECh. 15 - Prob. 13AAECh. 15 - Prob. 14AAECh. 15 - Prob. 15AAECh. 15 - Prob. 16AAECh. 15 - Prob. 17AAECh. 15 - Prob. 18AAECh. 15 - Prob. 19AAECh. 15 - Prob. 20AAECh. 15 - Prob. 21AAE
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Publisher:W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:9781337552516
Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning
01 - What Is an Integral in Calculus? Learn Calculus Integration and how to Solve Integrals.; Author: Math and Science;https://www.youtube.com/watch?v=BHRWArTFgTs;License: Standard YouTube License, CC-BY