Area Between Two Curves & FTC
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
Find the area enclosed between \(y=x^2\) and \(y=x+2\).
Find intersections
\(x^2=x+2\Rightarrow x^2-x-2=0\Rightarrow x=-1,2\)
Integrate upper minus lower
\(\displaystyle\int_{-1}^2[(x+2)-x^2]\,dx=[ frac{x^2}{2}+2x- frac{x^3}{3}]_{-1}^2\)
Evaluate
\((2+4- frac{8}{3})-( frac{1}{2}-2+ frac{1}{3})= frac{10}{3}+ frac{7}{6}= frac{27}{6}\)
Answer
\(\dfrac{9}{2}\) square units
Question 2
Find the area enclosed between \(y=\sin x\) and \(y=\cos x\) for \(0\leq x\leq\dfrac{\pi}{2}\).
Intersection
\(\sin x=\cos x\Rightarrow x= frac{\pi}{4}\)
Integrate
\(\displaystyle\int_0^{\pi/4}(\cos x-\sin x)\,dx+\int_{\pi/4}^{\pi/2}(\sin x-\cos x)\,dx\)
Evaluate each
\([\sin x+\cos x]_0^{\pi/4}+[-\cos x-\sin x]_{\pi/4}^{\pi/2}=(\sqrt2-1)+(\sqrt2-1)\)
Answer
\(2(\sqrt{2}-1)\) square units
Question 3
State the Fundamental Theorem of Calculus. Hence find \(\dfrac{d}{dx}\displaystyle\int_2^{x^2} e^t\,dt\).
FTC
If \(F(x)=\displaystyle\int_a^x f(t)\,dt\), then \(F'(x)=f(x)\)
Chain rule with upper limit \(x^2\)
\(\dfrac{d}{dx}\displaystyle\int_2^{x^2}e^t\,dt=e^{x^2}\cdot2x\)
Question 4
Find the area enclosed between \(y=e^x\) and \(y=e^{-x}\) for \(-1\leq x\leq1\).
Intersection at \(x=0\)
Both equal 1 there; \(e^x>e^{-x}\) for \(x>0\), so integrate by symmetry
Area
\(2\displaystyle\int_0^1(e^x-e^{-x})\,dx=2[e^x+e^{-x}]_0^1=2(e+e^{-1}-2)\)
Answer
\(2(e+e^{-1}-2)\) square units
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