The area under the curve of y=1/x from x=a to x=5 is approximately 0.916.

Question

The area under the curve of y=1/x from x=a to x=5 is approximately 0.916 where a is greater than or equal to 1 and less than 5. Find a using a calculator.

Answer 1

Evaluating the integral gives ln(5) - ln(a) = ln(5/a) = 0.916

taking both sides as exponents to e:

e^ln(5/a) = e^0.916

Using the log rule e^ln(x) = x

5/a = e^0.916

a = 5/e^0.916 = 5/2.499... ≈ 2.0

Answer 2

Since a > 1, the y = 1/x is positive between x = a and x = 5.

So, Area = ∫_{(from a to 5)} 1/x dx = lnx_{(from a to 5)} = ln5 - lna = 0.916

Therefore, lna = ln5 - 0.916 = 0.693437912

a = e^0.693437912 ≈ 2