# Integrals, Differential Equations and Limits

Please see the attached file for the fully formatted problems.

Question 1

Find

∫ x3+4

________________________________________x2 dx

Question 2

Solve the initial value problem:

dy

________________________________________dx = x^/¯(9+x2) ; y(-4) = 0

Question 3

Figure 3.1

f(x) = x2+3

Figure 3.2

ni=1 f(xi) x

Given Figure 3.1, find the exact area A of the region under f(x) on the

interval [1, 3] by first computing Figure 3.2 and then taking the limit as

n→∞

Question 4

Figure 4.1

limn sin 

________________________________________4 + sin 2

________________________________________n + sin 3

________________________________________n + sin n

________________________________________n

________________________________________n

Evaluate Figure 4.1 by interpreting it as the

limit of Riemann sums for a continuous function f defined on [0, 1].

Question 5

Figure 5.1

4∫-2(|3-|3x||)dx

Given Figure 5.1, first sketch the graph y = f(x) on the given interval.

Then find the integral of f using your knowledge of area formulas for

rectangles, triangles and circles.

Question 6

Suppose that a tank initially contains 2000 gal of water and the rate of

change of its volume after the tank drains for t min is V'(t) = (0.5)t - 30

(in gallons per minute). How much water does the tank contain after it

has been draining for 25 minutes?

Question 7

Evaluate

3∫1 6

________________________________________x2 dx

Question 8

Figure 8.1

1∫0 1

________________________________________x+2 dx

Find an upper and lower bound for the integral (Figure 8.1) using the

comparison properties of integrals.

Question 9

Apply the Fundamental Theorem of Calculus to find the derivative of:

h(x)= x∫2^/¯u-1dx

Question 10

Evaluate:

4∫1 (4+^/¯x)2

________________________________________2^/¯x dx

Question 11

Evaluate:

∫2cos2 xdx

Question 12

Figure 12.1

y = 9-x2 , y=5-3x

Sketch the region bounded by the graphs of Figure 12.1, and

then find its area.

Question 13

Figure 13.1

1?0x4dx

Approximate the integral (Figure 13.1); n=6, by:

a) first applying Simpsonfs Rule and

b) then applying the trapezoidal rule.

Question 14

Find the mass M (in grams) of a rod coinciding with the interval [0, 4]

which has the density function

(x)= 5 sin 

________________________________________4 x

Question 15

The region R is bounded by the graphs

x-2y=3 and x=y2

Set up(but do not evaluate) the integral that gives the volume of the solid

obtained by rotating R around the line x = -1.

Question 16

Figure 16.1

y=2x2

Find the volume of the solid that is generated by rotating the region

formed by the graphs of Figure 16.1 and y = 4x about the line x = 3.

Question 17

Use the method of cylindrical shells to find the volume of the solid

rotated about the line x = -1 given the conditions:

y = x3 - x2; y = 0; x = 0

Question 18

Find the length of the graph of

y = 1

________________________________________3 x3/2 - x1/2

from

(1, - 2

________________________________________3 )

to

(4, 2

________________________________________3 )

Question 19

A 10-ft trough filled with water has a semicircular cross section of

diameter 4 ft. How much work is done in pumping all the water over

the edge of the trough? Assume that water weighs

62.5lb/ft3

Question 20

Figure 20.1

y = x2

________________________________________2

The region in the first quadrant bounded by the graphs of y = x and Figure 20.1 is rotated around the line y = x. Find

(a) the centroid of the region and

(b) the volume of the solid of revolution.

#### Solution Summary

A variety of calculus problems are solved. The solution is detailed and well presented.