Determine the area of the shaded region in the following figur...

Question

Determine the area of the shaded region in the following figures.

Graph showing two curves with a shaded region between them, illustrating the area to be determined.

Accepted Answer

Find the area enclosed by the shaded region in the given figure. We notice here we're given two different graphs. X equals negative Y2 + 9 Y minus 6, and X equals Y2 + Y. We have 4 possible answers, being 18/5s, 8/3, 7/3, or 17/4ths in square units. Now, to solve this, we're going to use an integral, we first need to find our bounds. So To find the balance of our integral, we will set our equations equal to each other. We have negative Y2 + 9 Y minus 6 equals Y2 + Y. Let's go ahead and solve this. We can solve this here by adding Y2. Subtracting 9Y and adding 6 to the other side. We get 0 equals to westward. Minus 8 Y + 6. You can divide 2 out to get 0, equals Y2 minus 4 Y. Plus 3. And if we factor, we get two factors, Y minus 1 and Y minus 3. This gives us Y equals 1 and Y equals 3. These are the two bounds of our integral. Now, let's go ahead and create our integral. We have the integral from 1 to 3. And because we are in terms of DY, Our first equation will be our rightmost equation. And our second equation will be our leftmost equation. You will subtract these two equations. This equals from 1 to 3. Negative westward Plus 9 Y minus 6 minus Y2 plus Y D Y. Now we can simplify. We end up getting the interval. From 1 to 3 Of negative to Y2. Plus 81 Minus 6 DY. Now we can take our antiderivative. We end up getting the antiderivative, which is 2/3 Y cubed. Plus 4 5 squared. 61 from 1 to 3. And from here, we plug in our values. We get 2/3, multiplied by 3 cubed, plus 4 multiplied by 3 squared, minus 6 multiplied by 3. Minus in parenthesis. 2/3, multiplied by 1 cubed, plus 4 multiplied by +12 minus 6 multiplied by 1. We can now simplify everything. We end up getting a value of 0 minus. 8/3, which simplifies to just 8/3. Square units. If we look at our possible answers, we can determine then the answer to this problem is answer B. OK, I hope that help you solve the problem. Thank you for watching. Goodbye.

Determine the area of the shaded region in the following figures.

Watch next