Working with area functions Consider the func...

Question

Working with area functions Consider the function ƒ and the points a, b, and c.

(c) Evaluate A(b) and A(c). Interpret the results using the graphs of part (b) .

ƒ(𝓍) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says let GFX be equal to 8X multiplied by the quantity of X plus 2 in quantity, multiplied by the quantity of X minus 3 in quantity, and B equal to 5. Find the area function A of X, which is equal to the integral from 0 to X of GFTDT, then evaluate B. So first step is to find our area function, and we're given that our area function A of X is equal to the integral from 0 to X of GFT, and here we'll substitute in our expression for our function G. Which is going to be 8 T multiplied by the quantity of T + 2, and quantity multiplied by the quantity of T minus 3, and quantity DT. So, we need to simplify our integram. So, the first thing we want to do is multiply together our two binomial terms, but that's going to be equal to the integral from 0 to X of 8T. Multiplied by the quantity of T2 minus D minus 6 and quantity DT. And now we just distribute the AT and we see that this is going to be equal to the integral from 0 to X of the quantity of 8 E cubed. Minus 8 T squad minus 48 in quantity dT. So now we just need to integrate term by term. In doing so, we see that this is going to be equal to the quantity of 8 multiplied by, and here we're integrating T cube with respect to T, and that will give us the 4th divided by 4. Then we'll have minus 8, multiplied by, and here we're integrating T2d with respect to T, and that will give us the cubed divided by 3. Then we'll have minus 48, multiplied by, and here we're integrating T with respect to T and that'll give us T squared, divided by 2. In quantity, and this needs to be evaluated from 0 to X. So substituting in the limits of integration, we see that this is going to be equal to. And here for the first term, we'll have 8 divided by 4, which will give us 2, that's gonna be multiplied by X to the 4th, when we substitute in by for X T. For our upper limit, then we'll have minus 8 divided by 3, multiplied by X cubed. Then we'll have minus, and here we have 48 divided by 2, which will give us 24, and that gets multiplied by X squared. Then we'll have minus the quantity, and for our lower limit. We'll have 2 multiplied by 0 to the 4th. -8 divided by 3, multiplied by 0 cubed. Minus 24, multiplied by 0, squared and quantity. So now we just need to simplify this expression. So, everything for the lower bound is going to give us 0. And for the upper bound, we can factor out a 2 divided by 3 multiplied by X2. So we'll have 2 divided by 3, multiplied by X2. This gets multiplied by the quantity, and what's left is going to be 3 X2. -4 X. -36. In quantity. So this here is the answer that we're looking for for our area function. So, for the second part of this problem, we need to evaluate our area function at the value of B and we're told that B is equal to 5, so we're really looking for A of 5. So this is going to be equal to 2 divided by 3. Multiplied by 5 squared. Multiplied by the quantity of 3, multiplied by 5 squared. -4, multiplied by 5. Minus 36. In quantity. And when we evaluate this expression, we see that this is going to be equal to 950 divided by 3. So that is the answer that we're looking for for the second part of this problem. So I hope that this explanation has been useful, and I'll see you in the next video.

Working with area functions Consider the function ƒ and the points a, b, and c.
(c) Evaluate A(b) and A(c). Interpret the results using the graphs of part (b) .
ƒ(𝓍) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2

Key Concepts

Area Function

Definite Integral

Graphical Interpretation

Watch next