Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral.

Question

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. ​  ​

Quizplus · Accepted Answer

To sketch the region, we need to find the curve(s) that bound the area given by the integral. Looking at the integrand, we can rewrite it as:

$$f(x) = -x^2 + 4x -3$$

To find the x-intercepts of this function, we set $f(x) = 0$:

$$-x^2 + 4x -3 = 0 $$

Solving for x, we get:

$$x = \frac{4 \pm \sqrt{4^2 - 4(-1)(-3)}}{2(-1)} = 1, 3$$

Therefore, the region is bounded by the x-axis and the curve $f(x)$. This region is a trapezoid with height 3 and bases of length $f(1) = 0$ and $f(3) = -8$, so its area is:

$$A = \frac{1}{2}(0 + (-8))(3) = -12$$

Note that the area is negative, which is an indication that the function is below the x-axis. Therefore, the answer is:

Answer: D

Sketch the Region Whose Area Is Given by the Definite