Question

use the fundamental theorem of calculus to find the \area under curve\ of $f(x)=8x + 19$ between $x = 20$ and $x = 23$.

Explanation:

Step1: Find antiderivative of $f(x)$

The antiderivative of $f(x)=8x+19$ is $F(x)=4x^2+19x$ (using power rule: $\int x^n dx=\frac{x^{n+1}}{n+1}$ for $n
eq-1$, and $\int k dx=kx$ for constant $k$).

Step2: Apply Fundamental Theorem of Calculus

The area is $\int_{20}^{23} (8x+19)dx = F(23)-F(20)$.
First calculate $F(23)$:
$F(23)=4(23)^2+19(23)=4(529)+437=2116+437=2553$
Then calculate $F(20)$:
$F(20)=4(20)^2+19(20)=4(400)+380=1600+380=1980$

Step3: Compute the difference

$F(23)-F(20)=2553-1980=573$

Answer:

573