Question

Find the area bounded by the curve y=x2+3, the lines x=−2,x=3 and the x axis.

Answer 1

To find the area bounded by the curve y=x^2+3, the lines x=-2, x=3, and the x-axis, integrate the function y=x^2+3 from -2 to 3 and calculate the total area under the curve.

Step-by-step explanation:

To find the area bounded by the curve y=x^2+3, the lines x=-2, x=3, and the x-axis, you would need to set up an integral from -2 to 3 of the function y=x^2+3. This integral will give you the total area under the curve from x=-2 to x=3.

The integral is calculated as follows:

1. Integrate the function y=x^2+3 with respect to x from -2 to 3.
2. The antiderivative of x^2 is (1/3)x^3, and the antiderivative of 3 is 3x.
3. The definite integral from -2 to 3 of x^2 is ((1/3)×3^3 - (1/3)×(-2)^3), and for 3 it is (3×3 - 3×(-2)).
4. Add these results to get the total area.

This process finds the area under the curve and above the x-axis, thus giving the area occupied by the region we are interested in.