Is a constant of integration needed in a definite integral?

No; it cancels when the limits are subtracted.

What happens if you reverse the limits?

The sign of the integral changes.

When does a definite integral give the area under a curve?

When f(x)\geq0 between the limits.

What does a negative definite integral show?

The region lies below the x-axis.

How do you find the area of a region below the axis?

Take the positive value (the modulus) of the integral.

How do you find an area when the curve crosses the axis?

Find the roots, split the integral at them and add the positive areas.

What are the units of an area under a curve?

Square units, or the product of the axes' units.

What is the Fundamental Theorem of Calculus for definite integrals?

\inta^bf(x)\,dx=F(b)-F(a) where F'=f.

Definite integrals and area under a curveAQA A-Level Maths: Flashcards

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How do you evaluate $\int_a^bf(x)\,dx$?

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How do you evaluate $\int_a^bf(x)\,dx$ ?: $F(b)-F(a)$ , where $F$ is an integral of $f$ .
Is a constant of integration needed in a definite integral?: No; it cancels when the limits are subtracted.
What happens if you reverse the limits?: The sign of the integral changes.
What is $\int_a^af(x)\,dx$ ?: $0$
When does a definite integral give the area under a curve?: When $f(x)\geq0$ between the limits.
What does a negative definite integral show?: The region lies below the $x$ -axis.
How do you find the area of a region below the axis?: Take the positive value (the modulus) of the integral.
How do you find an area when the curve crosses the axis?: Find the roots, split the integral at them and add the positive areas.
What are the units of an area under a curve?: Square units, or the product of the axes' units.
Evaluate $\int_0^2 3x^2\,dx$ .: $\left[x^3\right]_0^2=8$
Evaluate $\int_1^4 x^{-\frac12}\,dx$ .: $\left[2\sqrt x\right]_1^4=2$
What is the Fundamental Theorem of Calculus for definite integrals?: $\int_a^bf(x)\,dx=F(b)-F(a)$ where $F'=f$ .

Exam questions on Definite integrals and area under a curve

The curve $C_1$ has equation $y=3x^2-2x+4$ .
Explain why $\int_1^3\left(3x^2-2x+4\right)dx$ is equal to the area of the region bounded by $C_1$ , the $x$ -axis and the lines $x=1$ and $x=3$ .2 marks
The curve $C_2$ has equation $y=(x-1)(x-5)$ . The finite region $R$ is bounded by $C_2$ and the $x$ -axis.
Find the area of the finite region bounded by $C_2$ , the coordinate axes and the line $x=1$ .2 marks
The curve $C_3$ has equation $y=4x-x^3$ .
Find the coordinates of the points where $C_3$ meets the $x$ -axis.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).