### Video Transcript

The table shows the values of a
function obtained from an experiment. Estimate the definite integral
between five and 17 of π of π₯ with respect to π₯ using three equal subintervals
with left endpoints.

Remember, we can estimate a
definite integral by using Riemann sums. In this case, weβre estimating the
integral between five and 17 of π of π₯. Now, it doesnβt really matter that
we donβt know what the function is. We have enough information in our
table to perform the left Riemann sum. The left Riemann sum involves
taking the heights of our rectangles as the function value at the left endpoint of
the subinterval. We want to use three equally sized
subintervals. So letβs recall the formula that
allows us to work out the size of each subinterval, in other words, the widths of
the rectangle.

Itβs Ξπ₯ equals π minus π over
π, where π and π are the endpoints of our interval and π is the number of
subintervals. In our case, weβre looking to
evaluate the definite integral between five and 17. So we let π be equal to five and
π be equal to 17. And we want three equal
subintervals. So weβll let π be equal to
three. Ξπ₯ is then 17 minus five all
divided by three, which is simply four. Then when writing a left Riemann
sum, we take values of π from zero to π minus one. Itβs the sum of Ξπ₯ times π of
π₯π for values of π from zero to π minus one. π₯π is π plus π lots of Ξπ₯. In this case, we know that π is
equal to five, and Ξ π₯ is equal to four. So our π₯π value is given by five
plus four π.

Well, since weβre using the left
Riemann sum, we begin by letting π be equal to zero. We need to work out π₯ zero. Itβs five plus four times zero,
which is simply five. We can find π of π₯ nought in our
table. Itβs negative three. Next, we let π be equal to
one. And we get π₯ one to be five plus
four times one, which is nine. We look up the value π₯ equals nine
in our table. And we see that π of nine is
negative 0.6. Next, we let π be equal to
two. And remember, weβre looking for
values of π up to π minus one. Well, three minus one is two. So this is the last value of π
weβre interested in. This time, thatβs five plus four
times two which is 13. We look up π₯ equals 13 in our
table. And we get that π of 13 and π of
π₯ two is 1.8.

Then, according to our summation
formula, we find the sum of the products of Ξπ₯ and these values of π of π₯π. And so, an estimate for our
definite integral is four times negative three plus four times negative 0.6 plus
four times 1.8, which is negative 7.2. An estimate for the definite
integral between five and 17 of π of π₯ with respect to π₯ using three equal
subintervals is negative 7.2.

Now, we donβt need to worry here
that our answer is negative. Remember, when weβre working with
Riemann sums, weβre looking at areas. But when the function values are
negative, the rectangle sits below the π₯-axis. And so, its area is subtracted.