Answer
$7.25$
Work Step by Step
Let us consider $R_n$ be the approximation obtained by using $n$ rectangles of same width.
The exact area can be found under the graph of $f$:
$A=\lim\limits_{n\to \infty} f(x_k)\triangle x$; where $\triangle x=\dfrac{b-a}{n}$
Given: $f(x)=4-x^2$
Here, we have $\triangle x=\dfrac{b-a}{n}=\dfrac{1-(-1)}{4}=0.5$
$A=\lim\limits_{n\to \infty} f(x_k)\triangle x$
or, $A=[f(x_1)+f(x_2)+f(x_3)+f(x_4)]\triangle x$
or, $A=[4-(-0.5)^2+4+0+4-(0.5)^2+4-1](0.5)$
Thus, $A=7.25$
