Mastering the Art of Integration: Vector-Valued Functions Explored
Integrating vector-valued functions involves computing the integral of functions whose outputs are vectors, typically represented as \(\int \mathbf{F}(t) dt\), where \(\mathbf{F}(t)\) could be a function from \(\mathbb{R}\) to \(\mathbb{R}^n\). This process requires taking the integral of each component function individually, often resulting in a vector of integrals. Applications include computing displacement from velocity in physics […]
How to Accurately Calculate the Area Between Polar Curves Using Integrals
To find the area between polar curves, identify the region bounded by two curves, \( r = f(\theta) \) and \( r = g(\theta) \), over an interval \([ \alpha, \beta ]\). The area between these curves is calculated by integrating the difference in their radial values squared: \([\text{Area} = \frac{1}{2} \int_{\alpha}^{\beta} \left( f(\theta)^2 – […]
