Single Variable Calculus: Early Transcendentals
8th Edition
ISBN: 9781305270336
Author: James Stewart
Publisher: Cengage Learning
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Chapter T, Problem 1DDT
(a)
To determine
To convert: The value
(b)
To determine
To convert: The value
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(3) (20 points) Let F(x, y, z) = (y, z, x²z). Define
E = {(x, y, z) | x² + y² ≤ z ≤ 1, x ≤ 0}.
(a) (2 points) Calculate the divergence V. F.
(b) (4 points) Let D = {(x, y) | x² + y² ≤ 1, x ≤ 0} Without calculation, show that
the triple integral
√ (V · F) dV = √ 2²(1.
= x²(1 − x² - y²) dA.
E
(2) (22 points) Let F(x, y, z) = (x sin y, cos y, ―xy).
(a) (2 points) Calculate V. F.
(b) (6 points) Given a vector field
is everywhere defined with V
G₁(x, y, z) = *
G2(x, y, z) = −
G3(x, y, z) = 0.
0
0
F(x, y, z) = (F₁(x, y, z), F₂(x, y, z), F(x, y, z)) that
F = 0, let G = (G1, G2, G3) where
F₂(x,
y,
y, t) dt
- √ F³(x, t, 0) dt,
*
F1(x,
y, t) dt,
t) dt - √ F
Calculate G for the vector field F(x, y, z) = (x sin y, cos y, -xy).
Evaluate the following integral over the Region R.
(Answer accurate to 2 decimal places).
√ √(x + y) A
R
R = {(x, y) | 25 < x² + y² ≤ 36, x < 0}
Hint: The integral and Region is defined in rectangular coordinates.
Chapter T Solutions
Single Variable Calculus: Early Transcendentals
Ch. T - Evaluate each expression without using a...Ch. T - Simplify each expression. Write your answer...Ch. T - Expand and simplify. (a) 3(x + 6) + 4(2x 5) (b)...Ch. T - Factor each expression. (a) 4x2 25 (b) 2x2 + 5x ...Ch. T - Simplify the rational expression. (a) x2+3x+2x2x2...Ch. T - Rationalize the expression and simplify. (a) 1052...Ch. T - Rewrite by completing the square. (a) x2 + x + 1...Ch. T - Solve the equation. (Find only the real...Ch. T - Solve each inequality. Write your answer using...Ch. T - State whether each equation is true or false. (a)...
Ch. T - Prob. 1BDTCh. T - Prob. 2BDTCh. T - Prob. 3BDTCh. T - Prob. 4BDTCh. T - Sketch the region in the xy-plane defined by the...Ch. T - FIGURE FOR PROBLEM 1 1. The graph of a function f...Ch. T - If f(x) = x3, evaluate the difference quotient...Ch. T - Find the domain of the function. (a)...Ch. T - How are graphs of the functions obtained from the...Ch. T - Without using a calculator, make a rough sketch of...Ch. T - Let f(x)={1x2ifx02x+1ifx0 (a) Evaluate f(2) and...Ch. T - If f(x) = x2 + 2x 1 and g(x) = 2x 3, find each...Ch. T - Prob. 1DDTCh. T - Prob. 2DDTCh. T - Find the length of an arc of a circle with radius...Ch. T - Prob. 4DDTCh. T - Express the lengths a and b in the figure in terms...Ch. T - If sinx=13 and secy=54, where x and y lie between...Ch. T - Prove the identities. (a) tan sin + cos = sec ...Ch. T - Find all values of x such that sin 2x = sin x and...Ch. T - Sketch the graph of the function y = 1 + sin 2x...
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- Find the volume of the solid that lies under the paraboloid z = 81 - x² - y² and within the cylinder (x − 1)² + y² = 1. A plot of an example of a similar solid is shown below. (Answer accurate to 2 decimal places). Volume using Double Integral Paraboloid & Cylinder -3 Hint: The integral and region is defined in polar coordinates.arrow_forwardEvaluate the following integral over the Region R. (Answer accurate to 2 decimal places). √4(1–2² 4(1 - x² - y²) dA R 3 R = {(r,0) | 0 ≤ r≤ 2,0π ≤0≤¼˜}. Hint: The integral is defined in rectangular coordinates. The Region is defined in polar coordinates.arrow_forwardEvaluate the following integral over the Region R. (Answer accurate to 2 decimal places). R - 1 · {(r,0) | 1 ≤ r≤ 5,½π≤ 0<1π}. Hint: Be sure to convert to Polar coordinates. Use the correct differential for Polar Coordinates.arrow_forward
- Evaluate the following integral over the Region R. (Answer accurate to 2 decimal places). √ √2(x+y) dA R R = {(x, y) | 4 < x² + y² < 25,0 < x} Hint: The integral and Region is defined in rectangular coordinates.arrow_forwardHW: The frame shown in the figure is pinned at A and C. Use moment distribution method, with and without modifications, to draw NFD, SFD, and BMD. B I I 40 kN/m A 3 m 4 marrow_forwardLet the region R be the area enclosed by the function f(x)= = 3x² and g(x) = 4x. If the region R is the base of a solid such that each cross section perpendicular to the x-axis is an isosceles right triangle with a leg in the region R, find the volume of the solid. You may use a calculator and round to the nearest thousandth. y 11 10 9 00 8 7 9 5 4 3 2 1 -1 -1 x 1 2arrow_forward
- Let the region R be the area enclosed by the function f(x) = ex — 1, the horizontal line y = -4 and the vertical lines x = 0 and x = 3. Find the volume of the solid generated when the region R is revolved about the line y = -4. You may use a calculator and round to the nearest thousandth. 20 15 10 5 y I I I | I + -1.5 -1 -0.5 0.5 1 1.5 2 2.5 3 -5 I -10 -15 I + I I T I I + -20 I + -25 I I I -30 I 3.5 4 xarrow_forwardplease show all the workarrow_forwardplease show all the workarrow_forward
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