Vault Of Euclid

Volume Under a Plane Over a Rectangle

Directional derivatives and the gradient warmup

Find the volume \( V \) under the plane \( z = 3x + 4y \) and over the rectangle \( \mathcal{R} \): \( 1 \leq x \leq 2 \), \( 0 \leq y \leq 3 \).

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What you need first

Basic integration of polynomials
Evaluating definite integrals
Understanding of functions of two variables

What this problem tests

Tests the ability to set up an iterated integral from geometric descriptions and perform partial integration.

This is a standard introductory problem in multivariable calculus for setting up and evaluating double integrals over rectangular regions.

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