Calculate 𝐿6 for 𝑓(𝑥)=√8x+3 over [1,3]. (Give your answer to three decimal places.)

Answer 1

I'm going to assume you meant √(8x + 3)

If you sketch it out you will see that this is nothing more than adding up the area of 6 rectangles:

The width of each rectangle is 1/3. To find that, just subtract the limits of integration (the 1 and the 3) and then divide by the number of rectangle (6): (3 - 1)/6 = 2/6 = 1/3.

The height of each rectangle is the function value of the left side. The first one is f(1), the second is f(4/3), the third f(5/3), then f(2) then f(7/3) then f(8/3):

f(1) = √(8(1) + 3) = √11 = 3.3166

f(4/3) = √(8(4/3) + 3) = √(41/3) = 3.6968

f(5/3) = √(8(5/3) + 3) = √(49/3) = 4.0415

f(2) = √(8(2) + 3) = √19 = 4.3589

f(7/3) = √(8(7/3) + 3) = √(65/3) = 4.6547

f(8/3) = √(8(8/3) + 3) = √(73/3) = 4.9329

So multiply each by the width 1/3 and add them all up:

3.6968(1/3) + 3.6968(1/3) + 4.0415(1/3) + 4.3589(1/3) + 4.6547(1/3) + 4.9329(1/3) = ??