WEBVTT
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The table shows the values of a function obtained from an experiment.
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Estimate the definite integral between five and 17 of π of π₯ with respect to π₯ using three equal subintervals with left endpoints.
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Remember, we can estimate a definite integral by using Riemann sums.
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In this case, weβre estimating the integral between five and 17 of π of π₯.
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Now, it doesnβt really matter that we donβt know what the function is.
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We have enough information in our table to perform the left Riemann sum.
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The left Riemann sum involves taking the heights of our rectangles as the function value at the left endpoint of the subinterval.
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We want to use three equally sized subintervals.
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So letβs recall the formula that allows us to work out the size of each subinterval, in other words, the widths of the rectangle.
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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.
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In our case, weβre looking to evaluate the definite integral between five and 17.
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So we let π be equal to five and π be equal to 17.
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And we want three equal subintervals.
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So weβll let π be equal to three.
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Ξπ₯ is then 17 minus five all divided by three, which is simply four.
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Then when writing a left Riemann sum, we take values of π from zero to π minus one.
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Itβs the sum of Ξπ₯ times π of π₯π for values of π from zero to π minus one.
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π₯π is π plus π lots of Ξπ₯.
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In this case, we know that π is equal to five, and Ξ π₯ is equal to four.
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So our π₯π value is given by five plus four π.
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Well, since weβre using the left Riemann sum, we begin by letting π be equal to zero.
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We need to work out π₯ zero.
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Itβs five plus four times zero, which is simply five.
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We can find π of π₯ nought in our table.
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Itβs negative three.
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Next, we let π be equal to one.
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And we get π₯ one to be five plus four times one, which is nine.
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We look up the value π₯ equals nine in our table.
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And we see that π of nine is negative 0.6.
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Next, we let π be equal to two.
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And remember, weβre looking for values of π up to π minus one.
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Well, three minus one is two.
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So this is the last value of π weβre interested in.
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This time, thatβs five plus four times two which is 13.
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We look up π₯ equals 13 in our table.
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And we get that π of 13 and π of π₯ two is 1.8.
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Then, according to our summation formula, we find the sum of the products of Ξπ₯ and these values of π of π₯π.
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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.
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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.
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Now, we donβt need to worry here that our answer is negative.
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Remember, when weβre working with Riemann sums, weβre looking at areas.
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But when the function values are negative, the rectangle sits below the π₯-axis.
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And so, its area is subtracted.