The square root is throwing me of doing the integration. Can someone please show the steps of integration.
Answer is $\$26.19$
Ok then can someone show me how we got the integration here?
calculus
The square root is throwing me of doing the integration. Can someone please show the steps of integration.
Answer is $\$26.19$
Ok then can someone show me how we got the integration here?
Best Answer
The height of the curve at $x = 50$ is $\sqrt{16 - 0.14\times50}=\sqrt{9}=3$. Thus, the consumer surplus will be given by $$ \int_0^{50}(p(x)-50)dx = \int_0^{50}\left[\sqrt{16 - 0.14x}-3\right]dx $$ As Tyler indicated, find this integral using the $u$-substitution of $u = 16 - 0.14 x$.
So, we want the function whose derivative is $\sqrt{16 - 0.14x}$. If we naively just integrated the outside function, we would get $\frac{2}{3} (16 - 0.14x)^{3/2}$. Let's check what the derivative of this function would be. We find, using the chain rule, $$ \frac d{dx} \frac{2}{3}(16 - 0.14x)^{3/2} = \sqrt{16 - 0.14 x} \cdot(-0.14) $$ This is almost what we want, except that we have a $-0.14$ multiplying at the end. So, we're going to take what we got originally and divide by $-0.14$. We then have $$ \frac d{dx} \frac{1}{-0.14}\frac{2}{3}(16 - 0.14x)^{3/2} = \frac{1}{-0.14}\sqrt{16 - 0.14 x} \cdot(-0.14) = \sqrt{16 - 0.14 x} $$ And that's the derivative we wanted! So, we now know that $$ \int \sqrt{16 - 0.14 x} = -\frac{1}{0.14}\frac{2}{3}(16 - 0.14x)^{3/2} + C $$ Use this to find the integral you need.