Double delight with 1729!!

Calculus Level 5

$\int_{-1729}^{1729} \int_{-14\pi}^{14\pi} (4\cos (y)+\sin^2 (x) + 2xy) \ dy \ dx$

Evaluate the above double integral. Round up your answer to the nearest integer.

1 solution

Peter Macgregor
Feb 19, 2015

Consider the inner integral first.

The $4\cos{y}$ contributes zero because it is integrated over an integer multiple of its period.

The $2xy$ contributes zero because $y$ is an odd function, and it is integrated over an interval centred on the origin.

Integrating the remaining term is easy because as far as the inner integral is concerned it is a constant.

Making use of these observations the double integral simplifies to

$\int_{-1729}^{1729} 28\pi\sin^2{x} \ dx$

Transform this into an integrable form with a trig. identity to get

$14\pi\int_{-1729}^{1729}1-\cos{2x}\ dx$

Now everyday integration gives, to the nearest integer, the answer $\boxed{152057}$

In the last step how did limits change from 1729 to 1792 In question its given 1729

Akhil D - 4 years, 7 months ago

sorry, it is a typo. I'll edit my solution.

Peter Macgregor - 4 years, 7 months ago

And I got answer as 152118 after rounding off

Akhil D - 4 years, 7 months ago