Mathematical Methods · Unit 4 · Further integration · Applications of integration
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A company models its daily profit over a 6-day period using the function \[ P(t) = 2t^2 - 8t + 6 \] where \( P \) is the profit in thousands of dollars and \( t \) is the time in days, \( 0 \leq t \leq 6 \). Calculate the total area enclosed between the profit curve and the \( t \)-axis over the domain \( 0 \leq t \leq 6 \).
A curve is defined by the function $f(x) = 12 - 3x^2$ for $0 \leq x \leq 2$. The region between the curve and the $x$-axis is shaded. Calculate the area of this shaded region, giving your answer correct to one decimal place.
The curve $y = 3\sin(x) + 2$ is defined over the domain $0 \leq x \leq \pi$. This curve lies entirely above the $x$-axis over this domain. (a) Write down a definite integral expression that represents the area enclosed between the curve and the $x$-axis. (1 mark) (b) Calculate this area, giving your answer correct to 2 decimal places. (2 marks)
Calculate the total enclosed area between the graph of \(y = x^3 - 4x\) and the x-axis from \(x = -2\) to \(x = 2\).
Calculate the area enclosed by the curve \(y = 4 - x^2\) and the \(x\)-axis between \(x = -1\) and \(x = 1\).