Video Transcript
Find the area under the curve 𝑦
equals negative 0.5𝑥 squared plus five from 𝑥 equals negative two to 𝑥 equals
one.
The area under the curve between
two points or, more specifically, the area between the curve and the 𝑥-axis can be
found by doing a definite integral between those two points. So we can find the area under the
curve between 𝑥 equals negative two and 𝑥 equals one by finding the integral
between negative two and one of negative 0.5𝑥 squared plus five with respect to
𝑥. And we recall that, to find the
integral of some 𝑎 times 𝑥 to the power of 𝑛 with respect to 𝑥, we find 𝑎 times
𝑥 to the power of 𝑛 plus one and then we divide by that 𝑛 plus one.
And of course we have that constant
of integration, 𝑐. This means the integral of negative
0.5𝑥 squared with respect to 𝑥 is negative 0.5𝑥 cubed over three. And the integral of five with
respect to 𝑥 is five 𝑥.
Remember, we’re going to be
evaluating this between negative two and one. And notice we haven’t included the
constant of integration here because we are actually evaluating between two
limits. We substitute 𝑥 is equal to one
and 𝑥 is equal to negative two into our expression for the integral. And then we find the
difference. So the area under the curve is
negative 0.5 times one cubed over three plus five times one minus negative 0.5 times
negative two cubed over three plus five times negative two.
Now at this point, we could
absolutely type this all into our calculator. But it’s really sensible to perform
an intermediate step. We’re going to evaluate the inside
of each pair of parentheses. And we see that the area is
twenty-nine sixths minus negative twenty-six thirds. That simplifies to 27 over two. And we see that the area under the
curve between 𝑥 equals negative two and 𝑥 equals one is 13.5 square units.
