Area functions Let A(x)

Question

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.

b. Find A(6).

ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3

Accepted Answer

Welcome back, everyone. Determine the area DFX enclosed by the x axis and the function Y equals GFX from X equals 0 to X equals 10, given that GFX is a piecewise function such that it is equal to x + 11 when x is less than or equal to 6, and it is equal to 5 when X is greater than 6. So let's understand what type of area we're trying to find. Now D of X is the area between two x values, starting with X1, which is 0. We want to finish with X2, which is equal to 10. Now if we look at the given graph, well, we start at the origin and then we finish at X equals 10. And now, well, if we look at the graph of the piece wise function, it consists of two regions. One is the region between 0 and 6 inclusive, right? Essentially, if we draw a vertical line, we can see that this is a trapezoid. And now the other region between 6 and 10, it is going to correspond to a rectangle, right? So we're looking for two areas. One of them is going to be a 1, the area of the trapezoid, and the second one is going to be a to the area of the rectangle. So now, let's begin with finding the area of the trapezoid. A1 is going to be equal to A + B. Divided by 2 multiplied by H, that's the formula for the area of a trapezoid. A and B, those are the bases. And now H is the height. We can say that H is simply the difference between the two x values. So that's X. 2 minus X1. We can choose different labels just to make sure that We don't confuse them with the original X1 and X2. Let's suppose that we are simply using X4 and X. We Now, what we have to notice is that in this specific notation, X4 is 6, that's the higher value. And X3 is 0. So we're subtracting 0 from 6, and this gives us a height of 6. Now, let's identify the a value, which is simply the Y coordinate at 0.0, right? So A equals J at 0.0. We're going to take the first part of the piece wise function. This gives us 0 + 11, which is 11, and the graph basically verifies that, right? So we have our a value. Now for the B value we can see that B corresponds to the value of the function when X is equal to 6. Graphically we can see that this must be 5. Let's check that. So we want to evaluate G at 0.6. Now we're taking the first part of the piece wise function once again, and this gives us -6 + 11, which is 5. So we have A, B, and we have the required height. Now, before we evaluate the area, A1, let's have a setup for a 2. Now A2 is a rectangle. So what we want to do is identify. The land in the west, right? So one of our parameters, or basically one of the sidelines required, is going to be the sideline between 6 and 10, right? So we can say that. Lent L. Is going to be equal to. 10 minus 6. Which is equal to 4. Now, for the other side length, we can immediately see that this is going to be equal to B. We already know that B is equal to 5. Right? So we are going to state that B is equal to 5. And now the total area is simply equal to A1 plus A2. We know that A1 is equal to A + B divided by 2 multiplied by H. And 82. Is going to be equal to B multiplied by L, right? Length multiplied by width. So B multiplied by L. and now all that we have to do is substitute the given values. Now, A. Is going to be equal to 11, B is equal to 5. We're going to divide that by 2 and multiply by the value of H. H is equal to 6. And then we're adding the product between B and L, the value of B is 5. And 1 is equal to 4. Let's simplify the result. We end up with 16 divided by 2, which is 88 multiplied by 6 gives us 48, so that's a 1. And a 2 is going to be 20. When we add these together, we get 68. So the total area is going to be equals to 68. Thank you for watching.

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.

b. Find A(6).

ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3 <IMAGE>

Key Concepts

Area Under a Curve

Piecewise Functions

Definite Integrals

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