Video Transcript
The function 𝑓 is continuous on
the closed interval negative four to four and satisfies the definite integral
between zero and four of 𝑓 of 𝑥 with respect to 𝑥 is equal to nine. Determine the definite integral
between zero and four of 𝑓 of 𝑥 minus six with respect to 𝑥.
To answer this question, we’ll
begin by recalling some basic properties of definite integrals. Firstly, we know that the integral
of the sum of two functions is equal to the sum of the integral of each respective
function. Similarly, the integral of the
difference of two functions is equal to the difference of the integrals of those
functions. This means we can begin by
rewriting our integral as the difference between the integral between zero and four
of 𝑓 of 𝑥 with respect to 𝑥 and the integral between zero and four of six with
respect to 𝑥. We also recall that for constant
function 𝑐, the definite integral between 𝑎 and 𝑏 of 𝑐 with respect to 𝑥 is
equal to 𝑐 times 𝑏 minus 𝑎.
Now, our 𝑐 is equal to six, and 𝑎
is zero, 𝑏 is four. So the definite integral between
zero and four of six with respect to 𝑥 is simply six times four minus zero, which
is equal to 24. As per the question, we now replace
the integral between zero and four of 𝑓 of 𝑥 with respect 𝑥 with nine. And we find that the integral of 𝑓
of 𝑥 minus six between the limits of zero and four with respect to 𝑥 is nine minus
24. That’s equal to negative 15. So, despite not knowing the
function 𝑓, we see that the definite integral between zero and four of 𝑓 of 𝑥
minus six with respect to 𝑥 is negative 15.
