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\text { (a) Consider the function } f(x)=-\sqrt{25-x^{2}} \text { . } i) Find the equation of the normal to the curve y = f(x) at the point with x-coordinate

a, where -5 < a < 5. ii) Show that this line passes through the origin. (b) i) Using the trapezoidal rule with 2n subintervals, find an expression for the ii) By taking the limit as n →oo, verify that taking more partitions leads to a better approximation. 1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\left(\frac{n(n+1)}{2}\right)^{2} \text { approximation to } \int_{0}^{2} x^{3} d x \text { in terms of } n \text { . You may find the following rule } helpful:

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