Computing the Definite Integral of a Polynomial

We want to focus on the definite integral of a polynomial function. These arise very commonly in calculus, so here are detailed solutions to two problems, one multiple-choice and one free-response, involving a definite integral of polynomial.

 

Free-Response Definite Integrals:

You will not commonly be asked to evaluate common definite integrals on the free-response, but rather you will be asked to find an area or compute a volume, which will require computing a common definite integral. Suppose we want to compute the volume of the solid obtained by revolving the function r(x) = -{1/6}(x - 6)(x+6) about the x-axis:

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The cross sections when cutting perpendicular to the x-axis are circles with radius given by the function r(x) = -{1/6}(x - 6)(x+6). The definite integral that needs to be evaluated is int{-6}{6}{pi.r(x)^2 dx},  since this is the area of a circle multiplied by the length of the interval from -6 to 6. We compute:

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Therefore to compute the integral we compute the sum of the integrals of the individual terms, since polynomials are sums of continuous functions:

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Recall the Fundamental Theorem of Calculus (FTC):

THEOREM: If v(x) is a continuous function with an antiderivative V(x), then V(b) - V(a) =int{a}{b}{v(x) dx}  where ,  are in the domain of v(x). 

The FTC says that we can pick any old antiderivative V(x) for v(x), so we need to compute a string of antiderivatives for the integrands of the terms in the sum. In the previous post we discussed but did not state:

The Power Rule: The derivative (x^n)‘=nx^n

We used this to find that the integral int{ }{ }{x^n dx} = {1/{n+1}}x^{n+1} + c, and since we only need one antiderivative to evaluate definite integrals, we can take  for use in this case.

Therefore we can evaluate (using the fact that int{ }{ }{x^4 dx} = {{x^5}/5}int{ }{ }{x^2 dx} = {{x^3}/3}int{ }{ }{x^0 dx} = int{ }{ }{1 dx} = {{x^1}/1} = x  and the FTC):

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You can use your calculator to get 723.823 units cubed.

 

Multiple-Choice Definite Integrals:

Here is a sample of a typical multiple-choice question asking for you to formulate a definite integral based on the same concept discussed above.

Question: A solid is generated by revolving the region enclosed by the function y = 2 sqrt{x}, and the lines x=2, x=3, y=1 about the x-axis. Which of the following definite integrals gives the volume of the solid? (Hint: Draw a picture)

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The idea for this problem is to recognize that this solid is a difference of integrals. Suppose that we had the volume of the function y=2 sqrt{x} when bounded by the lines  x = 2, x = 3, and rotated about the x-axis—then we would have the volume of the following solid:

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Given this volume, we would only need to subtract the volume of the following figure, derived by rotating y=1 bounded by x=2, x=3, about the x-axis:

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From the upper volume, with radius r_1 = 2sqrt{x}:

ctdioap_img8

Therefore we need to subtract the two integrals, however using the integral laws we can express this in the form pi.({r_1}^2 - {r_2}^2) dx, which we follow up by substitution of our names for r_1,r_2:

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So the answer is A.
To compute the value of the integral we see that
ctdioap_img10

This has the value 28.2743 units^2.

 

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