MATH 261 Los Angeles Pierce College Integration Problems Test Solve these 10 problems make sure to show all work. Label each question clearly. This is Calculus one so please don’t implement Calculus 2 in the work. The file has all the questions. Math 261 Test 4 Integration 1 and 2
Show all work. All numbered problems worth the same.
1) The graph of f(x) is the rate (gallons/minute) at which water is being
poured in or taken out of a container. Suppose there were 10 gallons of
water in the container at time t = 0 minutes.
a) Set up an expression that includes a definite integral that would give
the number of gallons in the container at t = 3 minutes, then compute
the expression and give the number of gallons.
b) Set up an expression that includes a definite integral that would give
the number of gallons in the container at t = 5 minutes, then compute
the expression and give the number of gallons.
2) Suppose a planet is actually growing in size (because aliens are injecting mass into the planet) such that an
object’s acceleration due to gravity on the planet is !”#$ % ‘#
()
*+,-
. (that’s -t not -1 ) . Suppose John
Carter jumps upward from the ground (0 ft. height) on this planet and his initial jump velocity is 8
()
/01
.
a) Find John Carter’s jump height function, h(t). Show all the steps leading to it.
b) Find the maximum height Carter could jump in this scenario. Give exact reduced fraction answer.
67
3) Given the definite integral 27 3 4 53 .
a) Estimate the integral using a Right Riemann sum with 5 sub-intervals. Show a graph with the
rectangles shown, and show the work – don’t just show the final answer!
b) Evaluate the integral all the way by hand to get the exact reduced fraction value of the integral.
4) Find the following indefinite and definite integrals showing all by hand steps all the way to the answer:
a) 27- 8 9,:*”;$ sin”?$ 5?
4
4
6
b) 2 √3 # ‘
53
!
$
6%$ 5) Given R is the region between (enclosed by) & % 3 4 !’5 & % 53 in the first quadrant.
a) Draw a picture and find the area of region R using calculus all the way to the exact answer by hand.
b) Rotate region R about the x –axis. Draw a picture of the solid created. Then show a representative
slab cut on the picture, and show the slab cut to the side of the picture and annotate appropriate
math on it in order to Just set up an integral that would find the resulting volume of the solid created.
Do NOT evaluate.
6) Rotate the region R in #6 above about the y –axis. Draw a picture of the solid created. Then just set up
an integral that would find the resulting volume of the solid created. Do NOT evaluate. (Shell method is
best here.)
7) Given F with F ′(x) = f(x) and F(0) = 1. The graph of f(x) is as shown.
a) Sketch a rough graph of F from 3 % 0 #* 3 % 3!
b) On what interval(s) is F increasing?
c) On what interval(s) is F concave down?
d) Where does F have a local minimum?
8) Copy the pyramid picture to your paper. The pyramid is to
be built up from the ground with stones weighing 2
pounds/cubic foot. Show illustrations and steps and write
a definite integral that would find the WORK required to
build the pyramid. Set up the integral then evaluate it
with technology – NOT by hand!
9) The region R is the region in the 1st quadrant enclosed by the y-axis, & % 3 4 , !’5 & % 8 ‘ 3 4 .
A solid is formed over the region that has square cross-sections perpendicular to the x-axis. The base of
each square cross-sections runs from & % 3 4 #* & % 8 ‘ 3 4 . Draw a 3-D picture of the solid and Just
set up but DO NOT evaluate a definite integral that finds the volume of the solid.
10) Free !
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