For the following exercises, evaluate the line integrals by applying Green’s theorem. 150. ∮ c ( − y d x + x d y ) , where C consists of line segment C 1 from (- 1, 0) to (1, 0), followed by the semicircular arc C, from (1,0) back to (1, 0)
For the following exercises, evaluate the line integrals by applying Green’s theorem. 150. ∮ c ( − y d x + x d y ) , where C consists of line segment C 1 from (- 1, 0) to (1, 0), followed by the semicircular arc C, from (1,0) back to (1, 0)
For the following exercises, evaluate the line integrals by applying Green’s theorem.
150.
∮
c
(
−
y
d
x
+
x
d
y
)
,
where C consists of line segment C1from (- 1, 0) to (1, 0), followed by the semicircular arc C, from (1,0) back to (1, 0)
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Velocity of a Ball Thrown into the Air The position function of an object moving along a straight line is given by s = f(t). The average velocity of
the object over the time interval [a, b] is the average rate of change of f over [a, b]; its (instantaneous) velocity at t = a is the rate of change of f at a.
A ball is thrown straight up with an initial velocity of 128 ft/sec, so that its height (in feet) after t sec is given by s = f(t) = 128t - 16t².
(a) What is the average velocity of the ball over the following time intervals?
[3,4]
[3, 3.5]
[3, 3.1]
ft/sec
ft/sec
ft/sec
(b) What is the instantaneous velocity at time t = 3?
ft/sec
(c) What is the instantaneous velocity at time t = 7?
ft/sec
Is the ball rising or falling at this time?
O rising
falling
(d) When will the ball hit the ground?
t =
sec
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Fundamental Theorem of Calculus 1 | Geometric Idea + Chain Rule Example; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=hAfpl8jLFOs;License: Standard YouTube License, CC-BY