Video Transcript
The function 𝑓 is continuous on
all real numbers. Write the integral from negative
two to three of 𝑓 of 𝑥 with respect to 𝑥 plus the integral from three to four of
𝑓 of 𝑥 with respect to 𝑥 minus the integral from negative two to zero of 𝑓 of 𝑥
with respect to 𝑥 in the form the integral from 𝑎 to 𝑏 of 𝑓 of 𝑥 with respect
to 𝑥.
We know that if capital 𝐹 of 𝑥 is
the antiderivative of 𝑓 of 𝑥, then the integral from a lower bound 𝑙 to an upper
bound 𝑢 of 𝑓 of 𝑥 with respect to 𝑥 is equal to evaluating our antiderivative at
𝑢 and subtracting this from our antiderivative evaluated at 𝑙. If the function 𝑓 is continuous
across its domain, then the integral from 𝑙 to 𝑢 of 𝑓 of 𝑥 with respect to 𝑥 is
equal to capital 𝐹 of 𝑢 minus capital 𝐹 of 𝑙. We can use this rule to rewrite the
three integrals given in this question.
Firstly, the integral from negative
two to three of 𝑓 of 𝑥 with respect to 𝑥 is equal to capital 𝐹 of three minus
capital 𝐹 of negative two. We can do the same with the
integral from three to four of 𝑓 of 𝑥 with respect to 𝑥. This is equal to capital 𝐹 of four
minus capital 𝐹 of three. Finally, we can write the last
integral from negative two to zero of 𝑓 of 𝑥 with respect to 𝑥 as capital 𝐹 of
zero minus capital 𝐹 of negative two.
Note that we need to subtract
this. This means that our expression is
equal to capital 𝐹 of three minus capital 𝐹 of negative two plus capital 𝐹 of
four minus capital 𝐹 of three minus capital 𝐹 of zero plus capital 𝐹 of negative
two. Next, we can simplify the
expression by canceling terms. We have capital 𝐹 of four minus
capital 𝐹 of zero. This means that the expression
given to us in the question is equal to evaluating the antiderivative of capital 𝐹
of four and subtracting the antiderivative of capital 𝐹 evaluated at zero.
Finally, as the function 𝑓 is
continuous for all real values and since capital 𝐹 is the antiderivative of 𝑓, we
can use the same rule to rewrite this expression as the integral from zero to four
of 𝑓 of 𝑥 with respect to 𝑥. The values of 𝑎 and 𝑏 are
therefore equal to zero and four, respectively.
