Question

Expressing geometric quantities with integrals: Express each of the given geometric quantities in terms of definite integrals. You do not have to solve the integrals.

The signed area of the region between the graph of$f\left(x\right)=\sin x$ and the x-axis on $\left[0,\frac{3\pi }{2}\right]$

Short answer

The integral will be${\int }_0^{\frac{3\mathrm{\pi }}{2}}\sin xdx.$

Step-by-step solution

Step 1. Given information.

The given function and interval is$f\left(x\right)=\sin xand\left[0,\frac{3\pi }{2}\right]$

Step 2. Required integral.

We know,

The signed area $={\int }_a^{b}f\left(x\right)dx$

Therefore, according to the question, the area can be represented as${\int }_0^{\frac{3\pi }{2}}\sin xdx.$