# 007B Sample Midterm 3

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This is a sample, and is meant to represent the material usually covered in Math 7B for the midterm. An actual test may or may not be similar.

Click on the  boxed problem numbers  to go to a solution.

## Problem 1

Divide the interval  ${\displaystyle [0,\pi ]}$  into four subintervals of equal length   ${\displaystyle {\frac {\pi }{4}}}$  and compute the right-endpoint Riemann sum of  ${\displaystyle y=\sin(x).}$

## Problem 2

Compute the following integrals:

(a)   ${\displaystyle \int x^{2}\sin(x^{3})~dx}$

(b)   ${\displaystyle \int _{-{\frac {\pi }{4}}}^{\frac {\pi }{4}}\cos ^{2}(x)\sin(x)~dx}$

## Problem 3

For a fish that starts life with a length of 1cm and has a maximum length of 30cm, the von Bertalanffy growth model predicts that the growth rate is  ${\displaystyle 29e^{-t}}$  cm/year where  ${\displaystyle t}$  is the age of the fish. What is the average length of the fish over its first five years?

## Problem 4

Find the volume of the solid obtained by rotating the region bounded by  ${\displaystyle y={\sqrt {\sin x}},~0\leq x\leq \pi ,}$  and  ${\displaystyle y=0}$  about the  ${\displaystyle x-}$axis. Sketch the graph of the region and a typical disk element.

## Problem 5

Evaluate the following integrals.

(a)   ${\displaystyle \int x\sin x~dx}$

(b)   ${\displaystyle \int {\frac {1}{(x-3)(x-2)}}~dx}$