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

A definite integral has limits: $\int_a^bf(x)\,dx=\left[F(x)\right]_a^b=F(b)-F(a)$, where $F$ is any integral of $f$. No constant of integration is needed because it cancels. This is the Fundamental Theorem of Calculus. Example: $\int_1^3(3x^2-2x+4)\,dx=\left[x^3-x^2+4x\right]_1^3=(27-9+12)-(1-1+4)=26$. Substitute the upper limit first, then subtract the value at the lower limit, and keep brackets round negative values.

• Reversing the limits changes the sign: $\int_b^af(x)\,dx=-\int_a^bf(x)\,dx$.
• $\int_a^af(x)\,dx=0$.
• A region can be split: $\int_a^cf(x)\,dx=\int_a^bf(x)\,dx+\int_b^cf(x)\,dx$. For $\int_3^1(3x^2-2x+4)\,dx$ the value is $-26$, the negative of $\int_1^3$. Before integrating, rewrite roots, products and fractions as sums of powers.

If $y=f(x)\geq0$ for $a\leq x\leq b$, the area between the curve, the $x$-axis and the lines $x=a$ and $x=b$ is $\int_a^bf(x)\,dx$. The integral gives the exact area because it is the limit of the sum of narrow rectangles, with $\int f(x)\,dx$ the reverse of differentiation. Check the curve is above the axis first: $3x^2-2x+4$ has discriminant $4-48<0$, so it is always positive, and the integral above, $26$, is the area in square units.

Where the curve is below the $x$-axis the integral is negative, but area is positive. For $y=(x-1)(x-5)$, $\int_1^5y\,dx=-\frac{32}{3}$ because the region is below the axis, so its area is $\frac{32}{3}$. Take the modulus of the integral for a region that lies entirely below the axis.

If the curve crosses the $x$-axis between the limits, integrating straight through lets positive and negative parts cancel. Find the roots, split at each one, find each area separately and add the positive values. For $y=6x-x^2$ from $x=0$ to $x=8$: $\int_0^6=36$ (above the axis) and $\int_6^8=-\frac{44}{3}$ (below), so the area is $36+\frac{44}{3}=\frac{152}{3}$, whereas $\int_0^8=\frac{64}{3}$ is only the net value. For an odd curve such as $y=4x-x^3$ the parts either side of the origin are equal in area but opposite in sign.