isi-entrance 2005 Q2

isi-entrance · India · solved Indefinite & Definite Integrals Piecewise/Periodic Function Integration
Let $f(x) = \int_0^1 |t - x|\, t\, dt$ for $x \in [0,1]$. Find $f(x)$ and sketch its graph.
$$f(x) = \int_0^x (x-t)t\,dt + \int_x^1 (t-x)t\,dt$$ $$= \left.\left(\frac{xt^2}{2} - \frac{t^3}{3}\right)\right|_0^x + \left.\left(\frac{t^3}{3} - \frac{xt^2}{2}\right)\right|_x^1$$ $$= x^3/3 - x/2 + 1/2$$ 
Let $f(x) = \int_0^1 |t - x|\, t\, dt$ for $x \in [0,1]$. Find $f(x)$ and sketch its graph.
Paper Questions
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