Definite integrals Use geometry (not Riemann sums) to evaluate...

Question

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.

∫₀⁴ ƒ(𝓍) d𝓍, where ƒ(𝓍) = {5 if 𝓍 ≤ 2

3𝓍 ― 1 if 𝓍 > 2

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to evaluate the definite integral using geometry, and we're given the integral from 2 to 6 of F of XDX, where F of X is equal to 6 if X is less than or equal to 4, and X + 2 if X is greater than 4. We're also given an empty graph below the problem. So, the first thing we want to do is to rewrite our integral, our integral from 0 to 6 of FXXDX by substituting in our function F of X. And since F of X is a piecewise function, we're going to end up with two separate intervals. So the interval from 2 to 6 of F of X. DX is going to be equal to the integral from 2 to 4 of 6 DX plus the integral from 4 to 6 of the quantity of X plus 2 in quantity DX. So, what we're going to need to do is to evaluate each of these two integrals separately using geometry. So, we're going to take a look at the first integral, that's gonna be the interval from 2 to 4 of 6 DX. And so what we're going to do is use our graph to draw this function on a graph. So, here we just have the function Y is equal to 6, so that is going to be a horizontal line on a graph between X equal to 2 and X equal to 4. And recall that the integral actually represents the area underneath the curve, so we need to find the area underneath that curve. And when we look at the shape that is made by that area, we see that we have a rectangle. So, this integral is just going to be the equal to the area of that rectangle. So, our integral here is going to be equal to, and herefore the area of a rectangle, a rectangle has a height of 6, and that gets multiplied by the width, which is the distance between X equal to 2 and X equal to 4, which is 2. And when we evaluate this expression, we see that our integral is going to be equal to 12. We'll also need to evaluate the interval from 4 to 6 of the quantity of X plus 2 in quantity DX. And again, we'll want to draw this function on our graph. So, here, X + 2 is going to give me a linear graph that starts when X is equal to 4. Our function here of X + 2 is going to be equal to 6, so we'll start at the point of 4.6, and go to the point of X equal to 6, and when X is equal to 6, X + 2 is equal to 8, so we'll go to the point of 6.8. And so we'll draw a line connecting those two points on our graph. And again, we need to find the area underneath this curve. And if we look at the area, its shape underneath the curve, we see that it is a trapezoid. So, here we just need to use the area of a trapezoid to evaluate this integral. And so this is going to be equal to, and here we call your formula for finding the area of a trapezoid. We will have one half, multiplied by the quantity. Here we have to sum up the lengths of the two parallel sides. So, on the left-hand side, that is parallel, We see that it has a length of 6. And then we'll have plus on the right hand side of our trapezoid is the other parallel side, and it has a length of 8, so we'll have the quantity of 6 plus 8. And then we need to multiply that by the height of our trapezoid, which is just going to be the distance from X equal to 4 to X equal to 6, which in this case is 2. And when we evaluate this expression, we see that this integral is going to be equal to 14. So that means that our integral, going from 2 to 6 of F of X. D X Is going to be equal to 12 + 14, which when we evaluate this expression is going to be equal to 26. So this here is the answer that we are looking for for this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.

∫₀⁴ ƒ(𝓍) d𝓍, where ƒ(𝓍) = {5 if 𝓍 ≤ 2
3𝓍 ― 1 if 𝓍 > 2

Key Concepts

Definite Integrals

Piecewise Functions

Geometric Interpretation of Integrals

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