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 a parabola and a line with a shaded area between them, illustrating the concept of area determination.

Accepted Answer

Welcome back, everyone. Find the area enclosed by the shaded region in the figure below A10/3, B 2/3, C, 20/3, and D 4/3. So let's recall that whenever we define a region between two curves, we basically can find the area enclosed by those curves as the integral. Of the difference between the two functions. We first will have to locate the upper curve. We can see from the image that the upper curve is the red one. It is above the shaded region, right? So we're going to take the equation of the upper curve, which is Y equals negative X. And we're going to subtract the equation of the lower curve. Our lower curve is the blue one. It basically defines our region from below, and it has an equation X2 minus 3X. So we want to subtract X2 minus 3X. And we're going to integrate with respect to x now because our integral must be a definite integral, we want to find the points of intersection. Let's notice that our curves intersect at X equals 0 at the origin. This is where the shaded region begins. And then our Shaded region ends at X equals 2, right? So essentially we are integrating from the lower limit of 0 up to the upper limit of 2. Meaning the area is the integral from 0 to 2 of minus X minus X2 minus 3 XD X. Well, then we have our setup. Now we can simplify this is equal to the integral from 0 to 2 of minus X minus X2 + 3 XD X. We're simply distributing the negative sign. And now we are ready to integrate the integral of negative axis, negative X squared divided by 2 according to the power rule. The integral of negative X cubed is negative. I'm sorry, the integral of negative X where this negative X cubed divided by 3, once again using the power rule. And the integral of 3 x is. 3 x squared divided by 2 once again using the Power rule and we want to evaluate the result from 0 to 2. We can also combine the like terms. That would be negative X2 + 3 X2 all divided by 2. Which is simply X2, and we're subtracting X cubed divided by 3. We're evaluating the result from 0 to 2. That's 2 squad minus 2 cubed divided by 3 minus 0 squad minus 0 cubed divided by 3. Which is equal to 4 minus 83, and that's 12/3 minus 8/3. That would be 4/3, which corresponds to the answer to the. Thank you for watching.

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

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