[Math] Calculating definite integrals from a graph consisting of two lines and a semicircle

Question

I honestly don't know how to do this I tried everything.

Answer 1

Think of $\int_a^b g(x)\,dx$ as the net area under the curve $y=g(x)$ from $x=a$ to $x=b$. For the first question you have $a=0$ and $b=10$. On that interval the graph of $y=g(x)$ is a straight line that drops from the point $\langle 0,20\rangle$ to the point $\langle 10,0\rangle$. The area under that ‘curve’ is just a triangle. What are the base and height of that triangle? What is its area? That area is $\int_0^{10}g(x)\,dx$.

For the second question you want $a=10$ and $b=30$. You’re supposed to assume that the graph on that interval is the lower half of a circle. What’s the radius of that semicircle? What’s the area between it and the $x$-axis? What’s the net area under the curve $y=g(x)$ on the interval from $x=10$ to $x=30$? Remember, the region is below the $x$-axis, and that does make a difference.

For the third question you have only to add your answers to the first two to get $\int_0^{30}g(x)\,dx$ and then add to that $\int_{30}^{35}g(x)\,dx$, which can be calculated just the way you calculate the first integral.