Use left end point approximation to approximate the area under the curve of 𝑓 of 𝑥 equals 𝑥 squared on the interval between zero and three; use subintervals with 𝑛 equals six.
So, we’re going to approximate the area under the curve of 𝑥 squared on the interval between zero and three. We have a really important tool which approximates the area under the curve using rectangles. And it’s called a Riemann sum. This means that we divide our interval into a number of subintervals. In this case, it’s six subintervals. And we create six rectangles which touch the curve. For this question, we’re asked to use a left end point approximation. So, this means that we want the top-left corner of each rectangle to touch the curve.
So, our approximation for the area under this curve is going to be the sum of the areas of these rectangles. So, we need to work out the area of each rectangle. Well, to work out the width of each rectangle, we work out the difference between zero and three, which is three. And we divide it by the number of rectangles, which is six. This gives us one-half. So, we can mark on the points on our 𝑥-axis. So, now let’s find the height of each rectangle.
Well, these points here are just the value of the function 𝑥 evaluated at the corresponding point on the 𝑥-axis. So, the height of the first subinterval is 𝑓 of zero, which is zero squared, which is zero. So, using the fact that the area of a rectangle is the height multiplied by the width, the area of the first subinterval is zero multiplied by a half, which is zero. We continue in the same fashion, finding the height of each rectangle by evaluating the function at each left-hand end point.
So, we find the height of the rectangles to be zero, one-quarter, one, nine over four, four, and 25 over four. So, the area of the second rectangle from the left is one-quarter multiplied by one-half, which is one over eight. For the next rectangle, the area is one multiplied by one-half, which is one-half. We continue in the same way, finding the area of each rectangle by multiplying the heights times the width. We then find the sum of all the areas of the rectangles by adding the areas together. And we find the total of all of these areas to be 55 over eight, which is our approximation for the area under the curve of 𝑥 squared between zero and three.