Question

AREA UNDER A CURVE - GEOMETRY

Answer 1

I also need an answer to this, pls help!!!

Answer 2

Recall that the area of a trapezoid is equal to the average of its bases times its height.

For the trapezoids shown here, each base corresponds to the value of
`y=2x^2` when
`x` is one of the endpoints of some interval, while the height is the length of that interval.

On the interval [-2, -1], the trapezoids has bases 2(-2)² = 8 and 2(-1)² = 2, and "height" -1 - (-2) = 1. Then its area is (8 + 2)/2 × 1 = 5.

On the interval [-1, 0], one of the bases is 2(-1)² = 2 and the other is 2(0)² = 0, and the height is again 0 - (-1) = 1. Then the trapezoid's/triangle's area is (2 + 0)/2 × 1 = 1.

Then the total area under the curve
`y=2x^2` on the interval [-2, 0] is approximately 5 + 1 = 6.

Compare this to the actual value of the area given by the definite integral,

`\displaystyle \int_(-2)^0 2x^2 \, dx = \frac{16}3 \approx 5.333\ldots`