Question

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${\int }_0^{\pi } \left({\sin }^{2}x+{\cos }^{2}x\right)dx$

Short answer

Ans: The exact value is,${\int }_0^{\pi } \left({\sin }^{2}x+{\cos }^{2}x\right)dx=\mathrm{\pi }.$

Step-by-step solution

Step 1. Given information.

given expression,

${\int }_0^{\pi } \left({\sin }^{2}x+{\cos }^{2}x\right)dx$

Step 2. The objective is to determine the exact value of the definite integral.  

The exact value is calculated as shown below,

$\begin{array}{r}{\int }_0^{\pi } \left({\sin }^{2}x+{\cos }^{2}x\right)dx\\ ={\int }_0^{\pi } 1dx\left[{\sin }^{2}x+{\cos }^{2}x=1\right]\\ =[x{]}_0^{\pi }\\ =\pi \end{array}$ Therefore, the exact value is $\mathrm{\pi }$.

Step 3. Check 

The required graph is,