Consider the definite integral J= _ 1 ^ 4 5t-4 ,dt

MATHEMATICS · Multi-part · CBSE · Mathematics · Integrals · Integrals

Problem

Consider the definite integral J=∫145t−4 dt.J=\int_{1}^{4}\sqrt{5t-4}\,dt. A reference book lists the antiderivative H(t)=215(5t−4)3/2H(t)=\frac{2}{15}(5t-4)^{3/2} for the integrand. (a) Differentiate H(t)H(t) and confirm algebraically whether H′(t)=5t−4H'(t)=\sqrt{5t-4} for all tt in the interval [1,4][1,4]. Clearly identify any constant factors that arise. (b) Provided the verification in part (a) is successful, evaluate JJ exactly using H(t)H(t) and the appropriate limits of integration. Show every substitution and simplification step. (c) Interpret the numerical value obtained in part (b) as the area of a region in the tt–yy plane. Describe that region verbally, specifying its horizontal boundaries and the curve that forms its upper boundary.

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