This problem is way easier if you approach it graphically, instead of solving it algebraically. Firstly, using one of the properties of integrals, you can separate the integral into two integrals like this:
If we analyze the first integral, since the function f(x) = x is a straight line, the area underneath f(x) between 0 and 1 is just a triangle like this:
The area enclosed is 0.5*1*1 = 0.5.
After that, if we analyze the second integral, we can first bring the constant 8 out of the integral. If you think about the function f(x) = sqrt(1-x^2), you would realize the area underneath the curve is a part of the circle. The area underneath f(x) between 0 and 1 would be a quarter of a circle that looks like this:
The area enclosed is 0.25*pi*r^2 = 0.25pi.
Now, we can just get these pieces together. Hence, the final answer to the integral would be 0.5 + 8 * 0.25 pi = 0.5 + 2pi.
Hi,
Thank you for your question.
This problem is way easier if you approach it graphically, instead of solving it algebraically. Firstly, using one of the properties of integrals, you can separate the integral into two integrals like this:
If we analyze the first integral, since the function f(x) = x is a straight line, the area underneath f(x) between 0 and 1 is just a triangle like this:
The area enclosed is 0.5*1*1 = 0.5.
After that, if we analyze the second integral, we can first bring the constant 8 out of the integral. If you think about the function f(x) = sqrt(1-x^2), you would realize the area underneath the curve is a part of the circle. The area underneath f(x) between 0 and 1 would be a quarter of a circle that looks like this:
The area enclosed is 0.25*pi*r^2 = 0.25pi.
Now, we can just get these pieces together. Hence, the final answer to the integral would be 0.5 + 8 * 0.25 pi = 0.5 + 2pi.
Does this answer your question?