A-level · Integration

Definite integration and area under a curveExam-style questions

Finding the area of a region with a curved edge. Thin strips and the limit of a sum; why an antiderivative measures area (the fundamental theorem); evaluating with square brackets, top limit minus bottom limit, with no + C; why a region below the x-axis counts as negative, so an integral and an area can differ; splitting and reversing the limits; and finding a limit that is a letter.

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5 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The diagram shows part of the curve $y = e^{-\sqrt{x}}$ .(a) Use rectangles of width $0.75$ to find upper and lower bounds for $\displaystyle\int_1^4 e^{-\sqrt{x}}\,dx$ . Give your answers correct to 3 significant figures.

(4 marks)
(b) (i) Use the substitution $t = \sqrt{x}$ to show that $\displaystyle\int_1^4 e^{-\sqrt{x}}\,dx = \displaystyle\int_1^2 2te^{-t}\,dt$ .

(ii) Hence show that $\displaystyle\int_1^4 e^{-\sqrt{x}}\,dx = 4e^{-1} - 6e^{-2}$ .

(7 marks)
Level 5 · 11 marksTry it