Estimate the integral between four and six of eight plus 𝑒 to the power 𝑥 with respect to 𝑥 using a Riemann sum with four left-hand rectangles.
Remember that this integral is the area between the curve eight plus 𝑒 to the power 𝑥 and the 𝑥-axis between the points four and six. Let’s call the function here 𝑓 of 𝑥. Riemann sums use rectangles to estimate the area under a curve. We’re going to use four rectangles.
We can work out the width of each of the rectangles by calculating the total width of the area, by subtracting four from six and then dividing by the number of rectangles we’re going to use, which is four. So the width of each rectangle is two over four, or a half or 0.5. You might recognise this as the formula Δ𝑥 equals 𝑏 minus 𝑎 over 𝑛, where Δ𝑥 is the width of each rectangle in the interval from 𝑎 to 𝑏 with 𝑛 rectangles. So we found our Δ𝑥 to be 0.5 the width of each rectangle.
Now we’ve been asked to use four left-hand rectangles. This means that the top left-hand corner of the rectangles touch the curve, as opposed to the right-hand corner of each triangle touching the curve. And so, for our left-hand Riemann sum, the height of each triangle is the value of the function at the left-hand point of each interval.
So for the first rectangle, the height is the function evaluated at the left-hand point, which is four. And to find the area of this rectangle, we multiply it by the width, which is Δ𝑥. So we’re going to find the sum of the areas of all of these rectangles. So for the second rectangle, its height is 𝑓 of 4.5 and the width is Δ𝑥. For the third triangle, its height is 𝑓 of five and width Δ𝑥. So we take the product of these. And for the fourth, its height is 𝑓 of 5.5 and width Δ𝑥. You probably recognise this as the Riemann sum formula. The sum between 𝑖 equals one and 𝑛 of 𝑓 of 𝑥𝑖 multiplied by Δ𝑥, where 𝑥 𝑖 are the left-hand endpoints.
So now we evaluate this sum. 𝑓 of four equals eight plus 𝑒 to the power four. And we multiply it by Δ𝑥, which is 0.5. We do this for all the terms in our sum. Note that here I’ve rounded the figures to three decimal places. But for full accuracy in the answer, you need to keep them to as many decimal places as you can. This gives us our final answer of 284.860.