(1) Consider the function

I(r)=0rex2dx,

where r0.

(a) Graph the function f(x)=ex2and indicate in your picture what the value I(r) represents. What do you think the value I() represents? We will try to compute it!

(b) Let's see if we can integrate it explicitly using one of the integration techniques we learned. First, try u-sub with u=x2. Explain why that's not useful.

(c) Now let's try integration by parts with u=ex2. Explain why that's not useful.

(d) Since our integration techniques were not very helpful, let's try to approximate the integral using rectangles. Based on the graph of f(x)=ex2, which kind of sample point should we use in our Riemann sum for a more accurate approximation: left endpoints, right endpoints or midpoints? Explain. Estimate I(1) using 4 rectangles with same base length and the type of sample point that will yield the best approximation.

(e) The computation you performed in (d) is probably not very accurate because yo used a small number of rectangles. If we use MATLAB, then we can increase the number of rectangles being used. Use our RIEMANN.m script (with left endpoints) to estimate:

(i) I(1) using 100 rectangles;

(ii) I(5) using 500 rectangles;

(iii) I(10) using 1000 rectangles;

(iv) I(100) using 10000 rectangles.

(f) If you did the routines right on MATLAB, you probably got roughly the same answer for subitems (iii) and (iv) in (e). Explain why that is the case using the graph you drew in (a).

(g) Use the estimates in (e) to obtain an estimate for I()=limrI(r).

(1) Consider the function where?\" 2 0. (a) Graph the function at) : 832 and indicate in your picture what the value IO") represents. What do you think the value 1(00) represents? We will try to compute it! (b) Let's see if we can integrate it explicitly using one of the integration techniques we learned. First, try u-sub with 'u, = a:2. Explain why that's not useful. 32 (c) Now let's try integration by parts with u = e- . Explain why that's not useful. (d) Since our integration techniques were not very helpful, let's try to approximate the 32 , which kind of sample point integral using rectangles. Based on the graph of at) = 6 should we use in our Riemann sum for a more accurate approximation: left endpoints, right endpoints or midpoints? Explain. Estimate HI) using 4 rectangles with same base length and the type of sample point that will yield the best approximation. (e) The computation you performed in (d) is probably not very accurate because yo used a small number of rectangles. If we use MATLAB, then we can increase the number of rectangles being used. Use our RIEMANNm script (with left endpoints) to estimate: (i) K1] using 100 rectangles; (ii) |(5) using 500 rectangles; (iii) K10) using 1000 rectangles; (iv) |(100) using 10000 rectangles. (f) If you did the routines right on MATLAB, you probably got roughly the same answer for subitems (iii) and (iv) in (e). Explain why that is the case using the graph you drew in (a). (g) Use the estimates in (e) to obtain an estimate for Koo) = lim 1(1"). T}w