Derive the formula for the area of a circle of radius r by evaluating an appropriate definite integral. (Hint: the equation y = square root {r^2 - x^2} gives a semicircle of radius r.) Calculate the integral either directly or using Fundamental Theo...
Example: Semicircle f(x) = √(1 − x2), from −1 to +1 A = π r2 / 2 = π / 2ConclusionWe can estimate the area under a curve by slicing a function upThere are many ways of finding the area of each slice such as: Left Rectangular Approximation Method (LRAM) Right ...
Evaluate the integral by interpreting it in terms of areas. {eq}\int_{-9}^{0} \; (1 + \sqrt{81 - x^2}) \, \mathrm{d}x {/eq} Integration: We have been given an integration and we have to solve it by areas. In this question the graph is of the ...
Use Green's theorem to evaluate closed integral yx^2 dx - x^2 dy where C is the boundary of the area bounded by a semicircle x = - square root of{25-y^2} and x=0. Use Green Theorem, to evaluate: surface integral (y+...
Evaluate the integral by changing to polar coordinates. Integral Integral_{D} e^{-x^2-y^2} dA, where D is the region bounded by the semicircle x = square root{4-y^2} and the y-axis. Calculate the triple integral \iiint_E y^2 z dV, where E is bo...