Find area with a reciprocal curve
A reciprocal curve can have a singularity and may lie above or below the horizontal axis. Check the interval and sign before integrating, then convert signed values to geometric area
\[A=\left|\int_a^b f(x)\,dx\right|\quad\text{if }f\text{ is continuous and has constant sign on }[a,b]\]
- Locate the singularity Set the denominator equal to zero and check whether the interval contains that input value \(\displaystyle x+1=0\Rightarrow x=-1,\quad [0,3]\text{ does not contain }-1\)
- Check the sign Determine whether the curve is above or below the horizontal axis on each part of the interval \(\displaystyle \frac{2}{x+1}>0\text{ on }[0,3]\Rightarrow A=\int_0^3\frac{2}{x+1}\,dx\)
- Use the logarithmic primitive A reciprocal linear function integrates to a logarithm with the inner coefficient divided out \(\displaystyle \int\frac{2}{x+1}\,dx=2\log_e\left|x+1\right|+C\)
- Split when the upper curve changes If another curve crosses the reciprocal curve, split at every intersection and use upper minus lower on each part \(\displaystyle A=\int_0^1\left(\frac{2}{x+1}-1\right)\,dx+\int_1^3\left(1-\frac{2}{x+1}\right)\,dx\)
Worked example
Find the exact geometric area between the reciprocal curve and the x-axis
\[f(x)=\frac{3}{1-x},\quad 2\le x\le4\]
- \(\displaystyle 1-x=0\Rightarrow x=1,\quad 1-x<0\text{ on }[2,4]\)The interval is on one side of the singularity and the curve is below the x-axis there
- \(\displaystyle \int_2^4\frac{3}{1-x}\,dx=\left[-3\log_e\left|1-x\right|\right]_2^4=-3\log_e(3)\)Evaluate the signed integral using the reciprocal logarithmic primitive
- \(\displaystyle A=-\left(-3\log_e(3)\right)=3\log_e(3)\)Negate the negative signed integral to obtain the positive geometric area
- \(\displaystyle A=3\log_e(3)>0\)The exact area is positive, as required for a geometric area
Common mistakes
Negating an above-axis integral
✕\(\displaystyle A=-\int_0^3\frac{2}{x+1}\,dx=-2\log_e(4)\)✓\(\displaystyle A=\int_0^3\frac{2}{x+1}\,dx=2\log_e(4)\)
The curve is above the x-axis, so its signed integral is already positive geometric areaIgnoring an intersection
✕\(\displaystyle A=\int_0^3\left(\frac{2}{x+1}-1\right)\,dx=2\log_e(4)-3\)✓\(\displaystyle A=\int_0^1\left(\frac{2}{x+1}-1\right)\,dx+\int_1^3\left(1-\frac{2}{x+1}\right)\,dx=1\)
The curves intersect at input value one, so the upper curve changes and the area must be split thereReversing upper and lower curves
✕\(\displaystyle \int_0^1\left(1-\frac{2}{x+1}\right)\,dx=1-2\log_e(2)\)✓\(\displaystyle \int_0^1\left(\frac{2}{x+1}-1\right)\,dx=2\log_e(2)-1\)
On [0,1], the reciprocal curve is above the line, so the integrand must be upper minus lowerDifficulty levels
introFind an exact area under a positive reciprocal curve on an interval that avoids its singularity
standardUse symmetry or a below-axis branch to find the exact geometric area
challengeFind area between a reciprocal curve and a line after locating both intersections
examChoose the correct area expression for a reciprocal curve and its line or axis boundary
Questions about this topic
Why must the singularity be checked first?+
The reciprocal curve is undefined at the denominator zero, so an interval containing that value cannot be treated as one ordinary definite integral
How is area found when the curve is below the x-axis?+
The signed integral is negative, so take its opposite, or take its absolute value, to obtain positive geometric area
Why is an area between two curves split at an intersection?+
The upper curve can change at an intersection. Splitting makes each integral use upper minus lower on its own subinterval
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Specialist Mathematics, chapter 5: Integral calculus
In this chapter
Integrate inverse circular kernels and change variablesAnti-differentiate reciprocal and linear-reciprocal functionsIntegration by recognition with logarithmic componentsAnti-differentiate by recognition from a supplied derivativeIntegrate using partial fractions, identities and partsDecompose rational expressions into partial fractionsCompound, double-angle and reciprocal identitiesAnti-differentiate exponential and circular functionsFind exact areas and assess numerical integration resultsFind the area between two curvesApproximate area by an endpoint or trapezium ruleFind areas involving reciprocal curvesAudit numerical integration loops and error enclosuresSet up volumes, parametric lengths and surfaces of revolution