Question

Find the area of the region under the graph of the function f on the interval
[0, 3].

f(x) = 3x − x^2

If you could show the steps, that would be most appreciated.

Answer 1

Answer:

9/2

Step-by-step explanation:

this is a simple integral function

given limits of interval [a,b] of a continuous function f(t), you can find the area under the curve by using:

$\int\limits^a_b {f(t)} \, dt$

$\int\limits^3_0 {3x-x^2} \, dx$

using the fundamental theorem of calculus that states the integral of f(x) in the interval [a,b] is  = g(a)-g(b), where g(x) is the antiderivative of f(x)

our g(x) = $\frac32x^2-\frac13x^3$

g(3)-g(0) = g(3) = 27/2 - 27/3 = 27/2-9 = 9/2