Browsing category integration

Spherical caps and coordinate systems

What is the volume above a plane, and inside a sphere of radius $r$, such that the radius of the circle where the two intersect is $R \sin(\alpha)$? What is this spherical sector's curved surface area? I've lost the precise wording of the question that drove a small cabal of

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How The Mathematical Pirate Integrates By Parts

The student sighed. "$\int x^3 e^{-2x} \dx$", he said. "That's going to be integration by parts. And it's going to take three steps. What a pain." "Aharr!" swashbuckled the Mathematical Pirate. "That's what you think!" "It is what I think," said the student, slightly bemused. "Would you like to know..."

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Ask Uncle Colin: A trigonometric integral

Dear Uncle Colin, I'm trying to find a definite integral: $\int_0^\pi \sin(kx) \sin(mx) \dx$, where $m$ and $k$ are positive integers and the answer needs to be simplified as far as possible. I've wound up with $\left[\frac{ (k+m) \sin((k-m) \pi) - (k-m)\sin((k+m)\pi) }{2(k-m)(k+m)}\right]$, but it's been marked wrong. -- Flat

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Ask Uncle Colin: Integration rules

Dear Uncle Colin, Why can't I work out $\int \left( \ln(x) \right)^2 \dx$ using the reverse chain rule? -- Previously Acceptable, Reasonable Technique Stumbles Hello, PARTS, There are two answers to this: the first is, you can't use the reverse chain rule -- which I learned as 'function-derivative' when I

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Where do the suvat equations come from?

Most of the suvat equations are pretty easy to derive, as soon as you realise acceleration ($a$, assumed constant) is the derivative of velocity ($v$) with respect to time, and velocity is the derivative of position ($s$), also with respect to time. For example: $ a = \diff{v}{t}$ $ \int_0^t

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Ask Uncle Colin: an integral that’s giving me a headache

Dear Uncle Colin, I've been trying to work out $I = \int_0^{\frac \pi 4} x \frac{\sin(x)}{\cos^3(x)} \d x$ for hours. It's the fifth time this week I've been up until the small hours working on integration and it's affecting my work and home life. I'm worried I'm becoming a calcoholic.

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There’s More Than One Way To Do It: Differential Equations

Until fairly recently, I had always done the kind of differential equations you see in Core 4 the same way: separate, integrate, substitute, celebrate. I have taught any number of students the dance; many of them have boogie-woogied their way into a correct answer in exams. But there's a variation

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Ask Uncle Colin: A Sticky Integral

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to and Uncle Colin will do what he can. Dear Uncle Colin, I've been trying to integrate $\int \frac{x^2}{x-1}

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Integrating $\sec^4(x)$

A student asks: How do you integrate $\int_{0}^{\frac{\pi}{4}} \sec^4(x) \d x$? Yuk. Let me say that again for good measure: yuk. That's going to need a trigonometric identity and, I think, a substitution. But that's ok: we can do that. Let's roll up our sleeves. Step 1: get rid of

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“A little biter of a question”

This problem came via the lovely @realityminus3 and caused me no end of problems - although I got there in the end. I thought it'd be useful to look at not just the answer, but the mistakes I made on the way. Maths is usually presented as 'here's what you

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