Area by geometry Use geometry to evaluate the...

Question

Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.

(b) ∫₆⁴ ƒ(𝓍) d𝓍

Graph of a function showing a piecewise linear shape, with labeled axes and a defined area under the curve.

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says the graph of H of X is given below, compute the interval from 7 to 4 of HFXDX using geometry. And below the problem we're given a graph of H of X, and H of X consists of four linear line segments. And we're also given 4 possible choices as our answers. For choice A, we have 5, for choice B, we have minus 20, for choice C, we have 10, and for choice D, we have minus 10. So we're asked to compute the interval from 7 to 4 of H of XDX. It's the first thing we want to do is rewrite this integral. So we'll have the interval from 7 to 4 of H of X. DX and this is going to be equal to, and what we want to do is switch the limits of integration. And recall that if we switch the limits of integration, we need to multiply our integral by -1. So this is going to be equal to minus the integral from 4 to 7 of HXDX. So, what we want to do now is to compute the integral from 4 to 7 of H of X. DX using geometry. And if we look at our graph, between X equal to 4 and X equal to 7, the area underneath that curve, which is what the integral represents, that that area actually forms a trapezoid. So, we're just going to use the area of a trapezoid to determine this integral. So, this integral, which is the interval from 4 to 7 of H of X and DX, is going to be equal to, and here we call your formula for the area of a trapezoid, that's going to be 12 multiplied by the quantity, and here we have to sum the sides, the lengths of the sides that are parallel to each other. And so one side occurs when Y is equal to 5, and that side that light segment occurs between X equal to 4 and X equal to 5. So that means it has a length of 1. So we'll have 1, and we'll have plus the other parallel side to our trapezoid occurs when Y is equal to 0, and it has a length that is between. X equal to 7 and X equal to 4, that means it has a length of 3. And then that quantity is going to be multiplied by the height of our trapezoid, and our height is just gonna be the distance from Y equal to 0 to Y equal to 5, which is just 5. And when we evaluate this expression, we see that it is going to be equal to 10. So that means that are integral. From 7 to 4 of HXDX. It's going to be equal to -10. So that is the answer that we're looking for for this problem. And if we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice D. So, I hope that this explanation has been useful, and I'll see you in the next video.

Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.
(b) ∫₆⁴ ƒ(𝓍) d𝓍

Key Concepts

Definite Integral

Area Under a Curve

Piecewise Function

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