Find the area of the region described in the following exercis...

Question

Find the area of the region described in the following exercises.

The region bounded by y=4x+4, y=6x+6, and x=4

Accepted Answer

Hello, in this video, we are going to be finding the area of the region bounded by the three given curves. Y is equal to 3X + 2, Y is equal to 5 X + 7, and X is equal to 3. Now, in order to approach this problem, let's go ahead and start by graphing the given curves. So, starting with the first curve, 3 X + 2, this is going to be a linear equation. With the slope of 3. Next, we have Y is equal to 5 X plus 7. For 5 X + 7, this is an equation of a line that has that starts at 7 on the y axis and has a slope of 5. And finally we have. A vertical asymptote or a vertical line, X is equal to 3. So this is a rough sketch of what the region, but trapped between the three curves is going to look like. That region is going to be the shaded red region here. Now, how do we find the area of this region? Well, in order to find the area, what we are going to do is we are going to integrate this area with respect to X. So, we are going to create a definite integral that is going to give us the area bounded between the two curves. Now, since we are integrating with respect to X, we need to first figure out the boundaries of integration. We know that the largest value of X that we have is going to be 3, but how do we find the smaller value for the lower boundary? Well, what we need to do is we need to find the value of X at the intersection of the two graphs. So in order to do this, we can take the two equations of both graphs, set them equal to each other, and solve for X. So doing that is going to give us 3 X + 2 is equal to 5 X plus 7. So now what we need to do is we need to solve for X. We will subtract 3 to 3 X to both sides of the equation, and subtract 7 to both sides of the equation. That will leave us with -5 is equal to to X, and dividing both sides of this equation by 2 is going to give us X is equal to -5 divided by 2. And so what this means is that the lower boundary is going to have a value of X equal to -5 divided by 2. So, the boundaries of integration are -5 divided by 2 to 3. Now, what we need to do is we need to set up the integral. Now, since we are integrating with respect to X, the area is going to be the definite integral from A to B, of the uppermost curve, subtract the lowermost curve. DX. Now, our uppermost curve in this case is going to be the line Y is equal to 5 X + 7, and the curve right below that is going to be 3 X + 2. So by plugging in our bounds and identifying our upper and lower curves, the integral that is going to give us the area of the shaded region is going to be the integral from -5 divided by 2 to 3 of the function 5 X plus 7. Minus 3X plus 2DX. Now what we need to do is we need to simplify this integral. So, starting with the simplification, we will distribute -1 into 3 X + 2. That will give us the integral from -5 halves to 3 of 5 X plus 7, minus 3X minus 2. By combining length terms, we can further simplify this integrand to be the integral of 2 X minus 9. Now we will take the antiderivative of both of the terms. If we take the antiderivative, we are left with X2 minus 9 X, and this is going to be evaluated from -5 halves to 3. Now, what we are going to do is we are going to plug in the values of the bounds by using the fundamental theorem of calculus part 2. That is going to leave us with the quantity 32 minus 9 multiplied by 3. Minus the quantity -5 halves squared minus 9 multiplied by -5 halves. After we algebraically simplify, this is going to simplify to leave us with 24 minus -25 divided by 4, and this will further simplify to give us 121 divided by 4. So the area between the three veg curves is going to be 121 divided by 4 square units, and with that being said, the solution to this problem is going to be C. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

Find the area of the region described in the following exercises.

The region bounded by y=4x+4, y=6x+6, and x=4

Key Concepts

Linear Equations

Area Between Curves

Definite Integrals

Watch next