Set up a sum of two integrals that equals the area of the shad...

Question

Set up a sum of two integrals that equals the area of the shaded region bounded by the graphs of the functions f and g on [a, c] (see figure). Assume the curves intersect at x=b.

Graph showing two curves, f(x) and g(x), with a shaded area between them from x=a to x=c, intersecting at x=b.

Accepted Answer

Welcome back, everyone. Given F of X and G of X are continuous functions on the interval from A to C inclusive, which of the integrals represents the area between F of X and GFX over the interval from A to C inclusive? We're given for answer choices. Those are different expressions of integrals, and we want to identify the correct answer. For this problem, let's recall that area between two curves is a definite integral. Between two values, A and B of F of X minus G of X. DX Now, in this definition, F of X is the top or the upper curve, and G of X is the bottom or the lower curve. And points A and B are the intersection points. So, what we want to do is simply analyze the given graph. We can see that between A and B. The upper curve is the red one, which is Y equals F of X. And then from B to C, the upper curve is the blue one, right? So, essentially our area is going to consist of two integrals. Now, the first one is going to be the integral from A to B. Of upper curve minus lower curve. So we take the red one, which is F of X. Minus the blue one which is G of X, and we integrate with respect to X. From here, we want to add the area from B to C. So we're going to add the integral from B to C of the upper curve, which is the blue one. So now we take G of X. And subtract. The lower curve, which is the red one, so we subtract F of X. And we integrate with respect to X. So we can now say that our area is the integral from A to B of F of X minus GFXDX plus the integral from B to C of GFX minus F of XDX. Considering the answer choices, we can conclude that the correct answer to this problem is C. Thank you for watching.

Set up a sum of two integrals that equals the area of the shaded region bounded by the graphs of the functions f and g on [a, c] (see figure). Assume the curves intersect at x=b.

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