# Calculus Review on Integrals

1. Evaluate the following indefinite integrals:

See attached

2. On a dark night in 1915, a German zeppelin bomber drifts menacingly over London. The men on the ground train a spotlight on the airship, which is traveling at 90 km/hour, and at a constant altitude of 1 km. The beam of the spotlight makes an angle θ with the ground.

1. Draw a diagram of the situation.

2. When the airship is 3 kilometers from the spotlight, how fast is θ changing?

3. An electrical circuit consists of two parallel resistors, with resistances R1 and R2 respectively. The total resistance R of the circuit (measured in Ohms Ω) is specified by

See attached

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#### Solution Preview

Please see attachments

In these questions, we use the following formula:

∫▒〖x^n dx=x^(n+1)/(n+1)+c〗

(1)∫▒〖1/x^2 dx=-1/x+c〗

(2)∫▒〖e^(-x) dx=-e^(-x)+c〗

(3)∫▒〖sin(t) cos(t)dt=∫▒〖sin(2t)/2 dt=sin(2t)/4+c〗〗

Assume x=cos(t), dx=-sintdt,

Therefore, ∫▒〖sin(t) cos(t)dt=∫▒〖-xdx=-x^2/2+c〗〗=-(cost)^2/2+c

(4)∫▒〖s^(1/2) ds=s^(1/2+1)/(1/2+1)+c=2/3 s^(3/2)+c〗

2.)

First, we find out the relationship between x and θ,

Then we consider x and θ are functions of t,

After this assumption, we could do derivative on both sides for equation: tan(θ)=1/x

Tan(θ)=1/x,sec^2〖θ*dθ/dt=-(1/x^2 dx)/dt〗

X=sqrt(3^2-1^2)=sqrt(8), dx/dt=90

Sec^2(θ)=1/(sqrt(8)/9)^2 =81/8

dθ/dt=-1/8*(81/8)*90=-113.91

1/R=1/R1+1/R2, when R1=80, R2=100, R=44.444

to find out the change rate for R,we consider ...

#### Solution Summary

The expert evaluates indefinite integrals in calculus. A diagram is drawn for a situation.