Questions: Find the area of the region bounded by the graphs of the given equations. y=3-x^2, y=3-6x

Transcript text: Find the area of the region bounded by the graphs of the given equations. \[ y=3-x^{2}, y=3-6 x \]

Solution

Solution Steps

Step 1: Find the Intersection Points

The given functions intersect at x-coordinates: [-3, 1]. These points serve as the limits of integration.

Step 2: Set Up the Integral

We determine which function is above the other in the interval [-3, 1] by evaluating them at a point between -3 and 1. Since the upper function is 3 - 6_x and the lower function is 3 - x^2, the area of the region is given by the integral of 3 - 6_x - 3 - x^2 over the interval [-3, 1].

Step 3: Calculate the Integral

The definite integral \(\int_{-3}^{1} (3 - 6*x - 3 - x^2) dx\) gives the area of the region bounded by the two curves. The calculated area is 33.33.