Consider the following.f(x) = 49 - x^{2} from x = 1 to x = 7, 4 subintervals(a) Approximate the

CoormaBak9 2020-11-14 Answered

Consider the following.
f(x)=49x2
from x=1 to x=7,4 subintervals
(a) Approximate the area under the curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the right-hand endpoints of the subintervals.
(b) Approximate the area under the curve by evaluating the function at the left-hand endpoints of the subintervals.

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Expert Answer

mhalmantus
Answered 2020-11-15 Author has 106 answers

(a) Given that, the function is f(x)=49x2 on the interval [1,7],n=4.
It is known that, the formula for the Riemann-sum is abf(x)dx=Δxi=0f(xi), where Δx=ban.
Obtain the value of Δx=ban
(a=1,b=7,n=4)
Δx=714
=64
=32
Divide the interval [1,7] into n=4 sub-intervals with length Δx=32 as [1,52],[52,4][4,112],[112,7],[112,7] and the right end-points are x1=52,x2=4,x3=112,x4=7. Find the area under the curve over [1,7] by using the right-end points as follows,
17(49x2)dxΔx(f(x1)+f(x2)+f(x3)+f(x4))
=32((49522)+(49(4)2)+(491122)+(49(7)2)
=32(42.75+33+18.75+0)
=32(94.5)
=141.75
Therefore, the area under the given curve over [1,7] by the right-end points approximation is 141.75.
(b) The left end-points are x0=1,x1=52,x2=4,x3=112.
Find the area under the curve over [1,7] by using the left-end points as follows,
17(49x2)dxΔx(f(x1)+f(x2)+f(x3)+f(x4))
=32(((49(1)2)+49(52)2)+(49(4)2)+(49(112)2)
=32(48+42.75+33+18.75)
=32(142.5)
=213.75
Therefore, the area under the given curve over [1,7] by the left-end points approximation is 213.75.

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