Q. 38 - Calculus

Question

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

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

Step-by-Step Solution

Answer

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

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,