Question

approximate the area under the graph of the function f(x)=15sinx from 0 to pi for n=4 and n=8 subitervals by using lower and upper sums

rap ec vals by ysing lae nd uppec sums lowec Sums a lo uppec sum se

Answer 1

(A)

(a)

f(z)d Az (f(zo)+f()2f (2)+.. +f()+f( where -a 0, b 4 We have that a T, n Therefore, Δχ 4 4 Divide the interval [O, π] into n 4 subintervals of length Δχ-- Now, we just evaluate the function at the left endpoints f (m) = f(a) = f (0) = 0-0 ) 10.6066017177982 f(5) f(m) 15-15 /h)-, (4)-부 = 10.6066017177982

Finally, just sum up the above values and multiply by 4 (0 +10.6066017177982 +15 +10.6066017177982)- 28.4417834690556 Answer: 28.4417834690556.

(b)

f(z)d Az (f(zo)+f()2f (r2)+... +f(n2)+f(xn 1)), where -a We have that a 0, b-TT, n-8 Therefore, Δz =--=- Divide the interval [O, π] into n 8 subintervals of length Δx = Now, we just evaluate the function at the left endpoints f (zo) f(a) f (0)-00 1))-17021485 ss +-= 5.74025148547635 10.6066017177982 (ī,--2-_ 13.8581929876693 +

f(q)=fG)-15-15 ) 10.6066017177982 f (z6 ) ร์ 15 " +-= 5.74025148547635

Finally, just sum up the above values and multiply by Дг (0 + 5.74025148547635 + 10.6066017177982+… + 10.6066017177982 + 5.74025148547635) = 29.6134740291833 Answer: 29.6134740291833.

(B)

(a)

1) + f(zn)), where Δ. b-a f(z)dz Δ2(f(x1) + f(x2) + f(z)+ +f(zn 0, b 4 We have that a π, n π_0 Therefore, ΔΖ Divide the interval [O, π] into n 4 subintervals of length Δ Now, we just evaluate the function at the right endpoints: 22-1 /h)-f (5) = 10.6066017177982 315y2 )-프 ,(n)-I( iO 606601717798 f(a)-f(b) f (r) = 0-0

Finally, just sum up the above values and multiply by (10.6066017177982 +15 +10.6066017177982 +0) 28.4417834690556 Answer: 28.4417834690556.

(b)

1) + f(zn)), where Δ. b-a f(z)dz Δ2(f(x1) + f(x2) + f(z) + +f(zn 0, b-T, n-8 We have that a Therefore, Ar Divide the interval [O, π] into n 8 subintervals of length ΔΖ 8'4 81 18 4 4 8 8 2288 Now, we just evaluate the function at the right endpoints: ,(z,-r()-15ý 을 +-= 5.74025148547635 ) 10.6066017177982 +-= 13.8581929876693 f()1515 13.8581929876693

3T 15V2 ) f (z6 )-f 10.6066017177982 " +-= 5.74025148547635 f (xs) f(b) = f (r) = 0 = 0

Finally, just sum up the above values and multiply by (5.74025148547635 10.606601717798213.8581929876693+... +5.74025148547635 +0) 29.6134740291833 Answer: 29.6134740291833.

[Please give me a Thumbs Up if you are satisfied with my answer. If you are not, please comment on it, so I can edit the answer. Thanks.]