### Video Transcript

Let π of π₯ equal two π₯ squared
plus three. Determine the area bounded by the
curve π¦ equals π of π₯, the π₯-axis, and the two lines π₯ equals negative one and
π₯ equals five.

Letβs begin with a sketch of the
region whose area weβre looking to calculate. Itβs bounded by a quadratic curve
with a positive leading coefficient and a π¦-intercept of three. Itβs also bounded by the two
vertical lines π₯ equals negative one and π₯ equals five and the π₯-axis. So itβs this area here that weβre
looking to find.

We recall that the area bounded by
the curve π¦ equals π of π₯, the π₯-axis, and the two vertical lines π₯ equals π
and π₯ equals π can be found by evaluating the definite integral from π to π of
π of π₯ with respect to π₯. Our function π of π₯ is two π₯
squared plus three. The lower limit for our integral,
the value of π, is the lower value of π₯. Thatβs negative one. And the upper limit, the value of
π, is the upper limit of π₯. Thatβs five. So the area weβre looking for is
equal to the integral from negative one to five of two π₯ squared plus three with
respect to π₯.

We recall that, in order to
integrate powers of π₯ not equal to negative one, we increase the power by one and
then divide by the new power. So the integral of two π₯ squared
is two π₯ cubed over three, and the integral of three is three π₯. We have then that the area is equal
to two π₯ cubed over three plus three π₯ evaluated between negative one and
five.

Remember, thereβs no need for a
constant of integration here, as this is a definite integral. We then substitute the limits,
giving two multiplied by five cubed over three plus three multiplied by five minus
two multiplied by negative one cubed over three plus three multiplied by negative
one. Thatβs 250 over three plus 15 minus
negative two-thirds minus three. That simplifies to 102. And so we can say that the area of
the region bounded by the curve π¦ equals π of π₯, the π₯-axis, and the two lines
π₯ equals negative one and π₯ equals five found by evaluating the definite integral
of our function π of π₯ between the limits of negative one and five is 102 square
units.