## Multivariate Calculus for Actuarial Science, Part I

The good news is that the double integrals on the actuarial exam do not get much more difficult than that. The bad news is that the numbers involved can get messier. Evaluate the integral

$$\int_1^2 \int_x^{x^2} \left(x^2 y^2 + 4\right) \, dy\ dx$$

carefully, then click on the box below to see the solution. You may need to concentrate to keep from making an addition error!

Show the solution

$$\int_1^2 \int_x^{x^2} \left(x^2 y^2 + 4\right) \, dy\ dx= \int_1^2 \left(\frac{1}{3}x^2 y^3 + 4y\right|_x^{x^2} dx= \int_1^2\left(\frac{1}{3}(x^8-x^5+12x^2-12x)\right)\, dx = \left(\frac{1}{54}(2x^9-3x^6+72x^3-108x^2)\right|_1^2=\frac{1013}{54}$$