Approximate the area under the curve

Question

Use the Midpoint Rule to approximate the area under the curve f(x)=x^2-3x+6 from x=4 to x=24. Use 5 rectangles to find your approximation. 

 

Answer= ---

Answer 1

Divide interval [4,24] into n=5 subintervals of length Δx=4 with the following endpoints: a=4,8,12,16,20,24=b.

Now, we just evaluate the function at those endpoints:

f(x0+x12)=f((4)+(8)2)=f(6)=24=24
f(x1+x22)=f((8)+(12)2)=f(10)=76=76
f(x2+x32)=f((12)+(16)2)=f(14)=160=160
f(x3+x42)=f((16)+(20)2)=f(18)=276=276
f(x4+x52)=f((20)+(24)2)=f(22)=424=424

 

Finally, just sum up the above values and multiply by Δx=4: 4(24+76+160+276+424)=3840
Answer: 3840.

 

As a check, the Anti-Derivative is (1/3)x^3 - (3/2)x^2 + 6x

 

(1/3)(24)(24)(24) - (3/2)(24)(24) + 6(24) - (1/3)(4)(4)(4) - (3/2)(4)(4) - 6(4)

 

3818 and 2/3