# 007B Sample Midterm 2

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

This problem has three parts:

(a) State both parts of the fundamental theorem of calculus.

(b) Compute   ${\displaystyle {\frac {d}{dx}}\int _{2}^{\cos(x)}\sin(t)~dt}$.

(c) Evaluate  ${\displaystyle \int _{0}^{\pi /4}\sec ^{2}x~dx}$.

## Problem 2

Evaluate

(a)   ${\displaystyle \int _{1}^{2}{\bigg (}2t+{\frac {3}{t^{2}}}{\bigg )}{\bigg (}4t^{2}-{\frac {5}{t}}{\bigg )}~dt}$

(b)   ${\displaystyle \int _{0}^{2}(x^{3}+x){\sqrt {x^{4}+2x^{2}+4}}~dx}$

## Problem 3

The population density of a plant species is  ${\displaystyle f(x)}$  individual per square meter, where  ${\displaystyle x}$  is the distance from the river, with  ${\displaystyle f(x)\geq 0}$  for  ${\displaystyle x\leq 100}$  and  ${\displaystyle f(x)=0}$  for  ${\displaystyle x\geq 100.}$ Construct a definite integral to calculate the number of plants along a section of the river of length  ${\displaystyle L.}$

## Problem 4

Find the area of the region bounded by  ${\displaystyle y=\ln x,~y=0,~x=1,}$  and  ${\displaystyle x=e.}$

## Problem 5

Evaluate the integral:

${\displaystyle \int {\frac {4x}{(x+1)(x^{2}+1)}}~dx}$