Find the area of the region under the graph of the function on the interval , using the fundamental theorem of calculus.Then verify your result using geometry.

Question

Find the area of the region under the graph of the function  on the interval  , using the fundamental theorem of calculus.Then verify your result using geometry.

Quizplus · Accepted Answer

The fundamental theorem of calculus states that if $f(x)$ is a continuous function on $[a,b]$, then the area of the region under the graph of $f(x)$ on the interval $[a,b]$ is given by $$\int_a^b f(x) dx.$$ Applying this theorem to the given function, we have $$\int_{-6}^{4} 3x^2-2x+1 \ dx = \left[x^3-x^2+x\right]_{-6}^{4} = (64-16+4)-(216-36-6) = \boxed{196}.$$ To verify this using geometry, we can graph the function and compute the area directly as the area of a trapezoid with base lengths 10 and 2 and height 16. This gives us an area of $\frac{(10+2)16}{2} = \boxed{96}$, which is not equal to the result we obtained using the fundamental theorem of calculus. Therefore, option B is the only correct choice.

Find the Area of the Region Under the Graph of the Function