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    Calculus Review on Integrals

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    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

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    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〗


    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

    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.