Video: APCALC04AB-P1B-Q32-153130927596

Using the right rectangular method and four subintervals of equal width, estimate ∫_0^8 |𝑓(π‘₯)| dπ‘₯, where 𝑓 is the function in the graph shown.

02:17

Video Transcript

Using the right rectangular method and four subintervals of equal width, estimate the definite integral between zero and eight of the absolute value of 𝑓 of π‘₯ with respect to π‘₯, where 𝑓 is the function in the graph shown.

The right rectangular method or the right Riemann sum is a way of approximating the area between a curve and the π‘₯-axis by splitting the region into rectangles. The formula that allows us to find the width of each rectangle, Ξ”π‘₯, is 𝑏 minus π‘Ž over 𝑛, where π‘Ž and 𝑏 are the lower and upper bounds of the region and 𝑛 is the number of subintervals. We also know that a definite integral is a way of calculating the exact area between the curve and the π‘₯-axis. And so, this means our right Riemann sum will allow us to estimate the definite integral between zero and eight of the absolute value of 𝑓 of π‘₯ with respect to π‘₯.

Our definite integral runs from π‘₯ equals zero to eight. So 𝑏 is eight and π‘Ž is zero. We want four subintervals. So we let 𝑛 be equal to four. Then, Ξ”π‘₯ is eight minus zero divided by four, which is simply two. And so, the width of each of our rectangles is two units. We’re working with the right rectangular method. So the height of each rectangle will be the value of the function at the right end of the interval. Since our interval starts at zero, the first rectangle looks a little something like this. Our second rectangle doesn’t really seem to exist. And that’s because the value of the function at the right side of this interval is zero. Our third rectangle is as shown. And this our fourth rectangle.

Now, we are going to need to be a little bit careful. And that’s because we’re actually finding the definite integral between zero and eight of the absolute value of the function. Geometrically, we graph an absolute value function by reflecting any part of the curve that is below the π‘₯-axis in the π‘₯-axis, as shown. And so, the definite integral between zero and eight of the absolute value of 𝑓 of π‘₯ with respect to π‘₯ will be the combined area of these rectangles.

The first rectangle has an area of two times two units and the second rectangle has an area of zero, as we saw. The third rectangle has an area of two times five units and the fourth rectangle is also two times two. That gives us a value of four plus 10 plus four, which is 18.

And so, an estimate of the definite integral between zero and eight of the absolute value of 𝑓 of π‘₯ with respect to π‘₯ is 18.

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