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If the Trapezoidal rule with single interval [0,1] is exact for approximating the integral $\displaystyle\int_0^1(x^3-c\ x^2) \ dx $,

then what is the value of c ?

I am confused with the word "exact". What does it mean ? and give me some hints to find c.

Thanks.

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"Exact" in this case means that the integration rule correctly calculates the value of the integral. Put another way, if $I$ is the exact value of the integral and $T$ is the approximation by the Trapezoid rule then $I-T = 0$.

To find $c$:

$$\int_{0}^{1} (x^3-cx^2) dx= \frac{1}{4}-\frac{c}{3}$$

Now using the trapezoid rule for a single interval gives the approximation: $$I \approx\frac{1}{2}(1-c)$$ But we know that the answer obtained from the trapezoid rule and the exact value of the integral should be the same. And so to find $c$ you need to solve

$$\frac{1}{4}-\frac{c}{3}=\frac{1}{2}(1-c)$$

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