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Approximating areas Estimate the area of the region bounded by the graph of $f(x)=x^{2}+2$ and the $x$-axis on [0,2] in the following ways.a. Divide [0,2] into $n=4$ subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.b. Divide [0,2] into $n=4$ subintervals and approximate the area of the region using a midpoint Riemann sum. Illustrate the solution geometrically.c. Divide [0,2] into $n=4$ subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

(A.)Left Riemann sum is $\dfrac{23}{4}$ = 5.75.(b)Midpoint Riemann sum is $\dfrac{53}{8}$ =6.625.(c) Right Riemann sum is$\dfrac{31}{4}$ =7.75.

Calculus 1 / AB

Calculus 2 / BC

Chapter 5

Integration

Section 1

Approximating Areas under Curves

Integrals

Integration Techniques

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All right. So this question, we are asked to estimate the area under the curve using the left green and some main point streaming some and and right Riemann, some methods. So let our function year after of X equals X squared, plus two number seven rolls of four. So the first step is gonna be to get the length of each step intervals. That's gonna be our range between zero and two. Divided by the number seven year olds. Full r equals 0.5. So next we have to figure out thank zero. So that's beginning us for seven role. That zero thanks. One is going to be eggs, Eero plus Delta X. So that's 0.5 and so on and so forth to get next to experience for the next step, we're gonna start with the left wing and some and we are going Teoh Um, yeah, I just use the form enough left even some so f of X, not plus fx. One does left next to the next three all that most blind by Delta X so ever. That's not well. That's the value of X squared post to when X equals zero. So That's two radio and X. Um, if x one, these value of X squared plus two one x one equals 0.5. So that's +212 and so on and so forth and multiply by Delta X being 0.5 and we get 5.75 And so this graph right here that's a sketch of the left with them left Freeman Very rough sketch of what the left or even sums look like. So next we're gonna do the midpoint calculation. Eso here, we're gonna have to use different Um, we have to use the midpoint, stow endpoints for X values and celebrate where we doesn't need those with a bar. So expire one is the halfway between zero and X one a k. It's the midpoint of the first interval. So expert one is 0.25 because it's not a zero and one is your wife. So then what we do is we plug those values onto our expression for the immediate point formula. So FX by her one, just like we did before. So that's the value. Its value. That's weird Post, too. When X equals 0125 So expired two is your 175 So f of X Part two is the value of X squared plus two one X equals zero points of five. And so you want to buy the that? Oh, you mad those things up By about zero point fire, you're gonna get 6.625 And this is a little sketch with the right Riemann sums, which are a press. They are an overestimate, as we can see. So I have the midpoint reading some. So that's gonna be our closest estimate. Lastly, you were going to do the right Riemann sons. So now we're using just our exes again on our X bars in the formula for the right Riemann some. So we go back a beer, remember? Experiment 0.5 f of X wanted X squared plus two one X equals your heart. Five. We were golden o and multiply by Point X and we're getting in our answer of 7.75

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