Math 152
Sample Exam 3

1.

Calculate the following integrals.

(a)

$\int_3^5 (4x-x^2) dx$

(b)

$\int_0^1 (3-x) dx$

(c)

$\int_6^{10}(x^4 + \frac{1}{x} - 4 + e^x)dx$

(d)

$\int_0^1 x(1-x^2)^3 dx$

(e)

$\int\frac {3x}{x^2-x-2}dx$

2.

Let A be the area between the graph of y = x² and the x-axis for $1 \leq x \leq 3$

(a)

Estimate A by computing $\Sigma f(x)\Delta x$ with $\Delta x = 0.5$

(b)

Determine A exactly by computing $\int_1^3 x^2 dx$

(c)

Are your answers from part a and b the same? Explain why or why not. (Hint: draw a picture)

3.

Given $f(x) = \sin[2(x-1)]$, what is the average value for f in $2 \leq x \leq 4$? Does this answer seem reasonable? (Hint: Sketch a graph of f(x).)

4.

A particle is moving with velocity v(t)=0.3+0.6 t, where time t is measured in seconds and distance is in meters.

(a)

Find the distance covered during first 5 seconds of movement.

(b)

Find the average velocity of the particle during first 5 seconds.

(c)

How much time is needed for the particle to move 100 meters?

(d)

Find the average velocity of the particle during the time calculated in (b).

5.

Calculate the volume of the solid formed by rotating the parabola y = x² + 3 about the x-axis for the interval [-2,2].

6.

Solve y' = -2y if y(1) = 3

7.

100 fruit flies are placed in a breeding container that can support a population of at most 5000 flies. The rate of growth of the population is directly proportional to the number present, and after 1 day, there are 102 flies in the container. How long will it take for the container to reach capacity?

8.

FInd the area between the graphs of the functions y = x³ and $y = \sqrt[3]{x}$.