Sketch the graph of the function 𝑥3:(a) Find the signed area between the curve, the 𝑥-axis, and the lines 𝑥=−2 and 𝑥=1.Signed area = (b) Find the geometric area between the curve, the 𝑥-axis, and the lines 𝑥=−2 and 𝑥=1.True area =

(a) To find the signed area between the curve, the x-axis, and the lines x=-2 and x=1, we need to integrate the function from -2 to 1. The signed area is the definite integral of the function from -2 to 1.

∫ from -2 to 1 of x^3 dx = [1/4 * x^4] from -2 to 1 = 1/4 * (1^4 - (-2)^4) = 1/4 * (1 - 16) = -15/4 = -3.75

So, the signed area is -3.75.

(b) The geometric area between the curve, the x-axis, and the lines x=-2 and x=1 is the absolute value of the signed area. This is because the geometric area is always positive, regardless of whether the function is above or below the x-axis.

True area = |-3.75| = 3.75

So, the geometric area is 3.75.