Please help with Definite Integrals

Question

The Riemann sum,  , is equivalent to  with  .

Write the integral that produces the same value as  .

options:

1)
2)
3)
4)

Answer 1

1,

∫baf(x)dx≈Δx(f(x0)+f(x1)+2f(x2)+...+f(xn−2)+f(xn−1)), where Δx=b−an

.

We have that a=0

, b=2, n=4

.

Therefore, Δx=(2−0)/4=1/2

.

Divide the interval [0,2]

into n=4 subintervals of length Δx=12: a=[0,12],[12,1],[1,32],[32,2]=b

Now, we just evaluate the function at the left endpoints:

f(x0)=f(a)=f(0)=3=3

f(x1)=f(12)=72=3.5

f(x2)=f(1)=4=4

f(x3)=f(32)=92=4.5

Finally, just sum up the above values and multiply by Δx=12

: 12(3+3.5+4+4.5)=7.5Answer: 7.5.

2.

∫baf(x)dx≈Δx(f(x0)+f(x1)+2f(x2)+...+f(xn−2)+f(xn−1)), where Δx=b−an

.

We have that a=0

, b=2, n=4

.

Therefore, Δx=(2−0/)4=1/2

.

Divide the interval [0,2]

into n=4 subintervals of length Δx=12: a=[0,12],[12,1],[1,32],[32,2]=b

.

Now, we just evaluate the function at the left endpoints:

f(x0)=f(a)=f(0)=2=2

f(x1)=f(12)=72=3.5

f(x2)=f(1)=5=5

f(x3)=f(32)=132=6.5

Finally, just sum up the above values and multiply by Δx=12

: 12(2+3.5+5+6.5)=8.5Answer: 8.5.

3.

∫baf(x)dx≈Δx(f(x0)+f(x1)+2f(x2)+...+f(xn−2)+f(xn−1)), where Δx=b−an

.

We have that a=0

, b=5, n=4

.

Therefore, Δx=(5−0)/4=5/4

.

Divide the interval [0,5]

into n=4 subintervals of length Δx=54: a=[0,54],[54,52],[52,154],[154,5]=b

.

Now, we just evaluate the function at the left endpoints:

f(x0)=f(a)=f(0)=0=0

f(x1)=f(54)=54=1.25

f(x2)=f(52)=52=2.5

f(x3)=f(154)=154=3.75

Finally, just sum up the above values and multiply by Δx=54

: 54(0+1.25+2.5+3.75)=9.375Answer: 9.375.

4.

∫baf(x)dx≈Δx(f(x0)+f(x1)+2f(x2)+...+f(xn−2)+f(xn−1)), where Δx=b−an

.

We have that a=2

, b=5, n=4

.

Therefore, Δx=(5−2)/4=3/4

.

Divide the interval [2,5]

into n=4 subintervals of length Δx=34: a=[2,114],[114,72],[72,174],[174,5]=b

.

Now, we just evaluate the function at the left endpoints:

f(x0)=f(a)=f(2)=2=2

f(x1)=f(114)=114=2.75

f(x2)=f(72)=72=3.5

f(x3)=f(174)=174=4.25

Finally, just sum up the above values and multiply by Δx=34

: 34(2+2.75+3.5+4.25)=9.375Answer: 9.375.