5.11 Definite integrals and areas questions for IB Maths AA

Show that $8g(x)=1-\cos 4x$.

Hence, find the exact value of ${\displaystyle {\int }_0^{\frac{\pi }{8}}g(x)dx}$, giving your answer in the form ${\displaystyle \frac{\pi -a}{b}}$, where $a,b\in {\mathbb{Z}}^{+}$.

Write down the equation of $L$.

Find the $x$-coordinate of P.

Find the area of the region enclosed by $L$ and the graph of $f$.

Find the acute angle between the tangent to the graph of $f$ at O and the $x$-axis.

The line $y=x\tan \theta$, where $\theta$ is acute, divides the region enclosed by the graph of $f$ and the $x$-axis into two parts of equal area.

Find the value of $\theta$.

Use your graphic display calculator to find the value of $I_1$, the value of $I_2$ and the value of $I_3$.

Suggest an exact value for $I_n$.

Show that $\sin ({\displaystyle \frac{\pi }{2}}-x)=\cos x$ and $\cos ({\displaystyle \frac{\pi }{2}}-x)=\sin x$.

By using the substitution $u={\displaystyle \frac{\pi }{2}}-x$, show that

Hence prove that your suggestion in part (b) is correct for all $n\in {\mathbb{Z}}^{+}$.

Now consider a function $f$ that is continuous for $0\le x\le a$, where $a>0$, and such that $f(x)+f(a-x)\mathrm{\ne }0$ for $0\le x\le a$.

Show that

Hence find the value of ${\displaystyle {\int }_0^4\frac{\sqrt{x}}{\sqrt{x}+\sqrt{4-x}}dx}$.

Now let $g$ be an even function that is continuous for $-a\le x\le a$.

Show that

Hence find the exact value of ${\displaystyle {\int }_{-\frac{\pi }{3}}^{\frac{\pi }{3}}\frac{\cos x}{1+e^x}dx}$.

Show that $f(x)=1+\cos 2x+\sin 2x$.

Find the $x$-coordinates of the stationary points on the graph of $f$.

Show that the local maximum value of $f$ is $1+\sqrt{2}$.

By writing $f(x)=2\cos x(\cos x+\sin x)$, find the $x$-intercepts of the graph of $f$.

The region $R$ is enclosed by the graph of $f$, the $x$-axis and the $y$-axis. Find the exact area of $R$.

Show that the graphs of $f$ and $g$ meet where $\cos x(2\sin x-1)=0$, and hence find the exact $x$-coordinates of the points where they meet.

Solve the inequality $f(x)>g(x)$.

Between $x={\displaystyle \frac{\pi }{6}}$ and $x={\displaystyle \frac{\pi }{2}}$, the graphs of $f$ and $g$ enclose a region. Find its area.

Find the total area of the regions enclosed by the graphs of $f$ and $g$ between $x={\displaystyle \frac{\pi }{6}}$ and $x={\displaystyle \frac{5\pi }{6}}$.

For ${\displaystyle \frac{\pi }{6}}<x<{\displaystyle \frac{\pi }{2}}$, the vertical distance between the graphs of $f$ and $g$ is $d(x)$.

Find the maximum value of $d(x)$.

Find the exact value of $f\left(\frac{\pi }{6}\right)$.

Find the exact value of $f\left(\frac{5\pi }{6}\right)$.

Find the exact value of $f\left(\frac{7\pi }{6}\right)$.

Use your GDC to sketch the graph of $y=f(x)$ for $-\pi \le x\le 2\pi$. Clearly indicate the coordinates of any maximum and minimum points.

Show that $f(x)=\pi -x$ for $\frac{\pi }{2}\le x\le \frac{3\pi }{2}$.

Write down the period of $f$.

Hence write down an expression for $f(x)$ for $\frac{3\pi }{2}\le x\le \frac{5\pi }{2}$.

Show that $f$ is an odd function.

Write down an expression for $g(x)$ for $0\le x\le \pi$.

Show that $g(x)=2\pi -x$ for $\pi \le x\le 2\pi$.

Let $h(x)=f(x)+g(x)$, for $0\le x\le 2\pi$.

Find an expression for $h(x)$ on each of the intervals $0\le x\le \frac{\pi }{2}$, $\frac{\pi }{2}\le x\le \pi$, $\pi \le x\le \frac{3\pi }{2}$ and $\frac{3\pi }{2}\le x\le 2\pi$.

Find all the solutions of the equation $f(x)={\displaystyle \frac{x}{2}}$, for $x\ge 0$.

Show that, for $\cos x\mathrm{\ne }0$, $f^{\mathrm{\prime }}(x)={\displaystyle \frac{\cos x}{\mathrm{\mid }\cos x\mathrm{\mid }}}$.

Hence explain why $f$ is not differentiable at $x=\frac{\pi }{2}$.

Find the exact value of ${\displaystyle {\int }_0^{10\pi }\mathrm{\mid }f(x)\mathrm{\mid }dx}$.