DP Mathematics HL Questionbank

• Syllabus

6.5

Sub sections and their related questions

Anti-differentiation with a boundary condition to determine the constant of integration.

• 16M.1.hl.TZ2.3b: Hence find...
• 16M.2.hl.TZ2.12a: (i)     Show that...
• 17M.1.hl.TZ1.11d: Hence find the value of $p$ if $\int_0^1 {f(x){\text{d}}x = \ln (p)}$.
• 18M.1.hl.TZ2.6b: Hence, or otherwise, find...

Definite integrals.

• 09M.1.hl.TZ1.9: (a)     Let $a > 0$ . Draw the graph of $y = \left| {x - \frac{a}{2}} \right|$ for...
• 09M.1.hl.TZ2.4: (a)     Show that \(\frac{3}{{x + 1}} + \frac{2}{{x + 3}} = \frac{{5x + 11}}{{{x^2} + 4x +...
• SPNone.3ca.hl.TZ0.4b: Determine the value of $\int_{ - a}^a {f(x){\text{d}}x}$ where $a > 0$ .
• 13M.1.hl.TZ1.10b: Find \(\int_{\frac{1}{{n + 1}}}^{\frac{1}{n}} {\pi {x^{ - 2}}\sin (\pi {x^{ - 1}}){\text{d}}x}...
• 13M.1.hl.TZ1.10c: Evaluate $\int_{0.1}^1 {\left| {\pi {x^{ - 2}}\sin (\pi {x^{ - 1}})} \right|{\text{d}}x}$.
• 13M.1.hl.TZ2.1: Find the exact value of...
• 14M.1.hl.TZ1.5c: Hence find the value of...
• 13N.1.hl.TZ0.12f: S is rotated through $2\pi$ radians about the x-axis. Find the value of the volume generated.
• 13N.1.hl.TZ0.12e: Hence find the value of $\int_0^{\frac{\pi }{2}} {{{\cos }^6}\theta {\text{d}}\theta }$.
• 15N.2.hl.TZ0.5a: (i)     Express the area of the region $R$ as an integral with respect to $y$. (ii)    ...
• 16M.1.hl.TZ2.3b: Hence find...
• 17M.1.hl.TZ1.11d: Hence find the value of $p$ if $\int_0^1 {f(x){\text{d}}x = \ln (p)}$.
• 18M.1.hl.TZ1.4a: $\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x}$.
• 18M.1.hl.TZ1.4b: $\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x}$.
• 18M.1.hl.TZ1.7b: Find $\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x}$.
• 18M.2.hl.TZ1.9d: The shaded region is rotated by 2$\pi$ about the $y$-axis. Find the volume of the solid formed.
• 18M.1.hl.TZ2.6b: Hence, or otherwise, find...

Area of the region enclosed by a curve and the $x$-axis or $y$-axis in a given interval; areas of regions enclosed by curves.

• 12M.1.hl.TZ1.2b: Hence state the value of (i)     $f'( - 3)$; (ii)     $f'(2.7)$; (iii)  ...
• 12M.1.hl.TZ1.6a: Find the area of region A in terms of k .
• 12M.1.hl.TZ1.6b: Find the area of region B in terms of k .
• 12M.1.hl.TZ1.6c: Find the ratio of the area of region A to the area of region B .
• 12M.1.hl.TZ2.10c: The region bounded by the graph, the x-axis and the y-axis is denoted by A and the region bounded...
• 12N.2.hl.TZ0.9: Find the area of the region enclosed by the curves $y = {x^3}$ and $x = {y^2} - 3$...
• 08M.1.hl.TZ1.6: Find the area between the curves $y = 2 + x - {x^2}{\text{ and }}y = 2 - 3x + {x^2}$ .
• 08M.2.hl.TZ2.3: The curve $y = {{\text{e}}^{ - x}} - x + 1$ intersects the x-axis at P. (a)     Find the...
• 11M.1.hl.TZ2.13a: (i)     Sketch the graphs of $y = \sin x$ and $y = \sin 2x$ , on the same set of axes, for...
• 11M.1.hl.TZ2.13c: The increasing function f satisfies $f(0) = 0$ and $f(a) = b$ , where $a > 0$ and...
• 09M.1.hl.TZ2.11: A function is defined as $f(x) = k\sqrt x$, with $k > 0$ and $x \geqslant 0$ . (a)    ...
• 10M.2.hl.TZ1.10: The diagram below shows the graphs of $y = \left| {\frac{3}{2}x - 3} \right|,{\text{ }}y = 3$...
• 10M.2.hl.TZ2.11: The function f is defined...
• 10N.1.hl.TZ0.13: Consider the curve $y = x{{\text{e}}^x}$ and the line \(y = kx,{\text{ }}k \in...
• 13M.2.hl.TZ2.12b: A different solution of the differential equation, satisfying y = 2 when $x = \frac{\pi }{4}$,...
• 11N.1.hl.TZ0.7: The graphs of $f(x) = - {x^2} + 2$ and $g(x) = {x^3} - {x^2} - bx + 2,{\text{ }}b > 0$,...
• 11N.2.hl.TZ0.1b: Find the area of the region bounded by the graph and the x and y axes.
• 11M.1.hl.TZ1.7: Find the area enclosed by the curve $y = \arctan x$ , the x-axis and the line $x = \sqrt 3$ .
• 14M.1.hl.TZ1.6: The first set of axes below shows the graph of $y = {\text{ }}f(x)$ for...
• 14M.1.hl.TZ1.11e: The graph of $y = {\text{ }}f(x)$ crosses the $x$-axis at the point A. Find the area...
• 14M.1.hl.TZ2.13e: Find the area of the shaded region. Express your answer in the form...
• 14M.2.hl.TZ2.3b: Find the area enclosed between the two graphs for...
• 13N.1.hl.TZ0.10f: Find an exact value for the area of the region bounded by the curve $y = g(x)$, the x-axis and...
• 14N.1.hl.TZ0.11d: A region $R$ is bounded by the graphs of $y = g(x)$, the tangent $T$ and the line...
• 15M.1.hl.TZ2.11d: Show that the area bounded by the graph of $y = g \circ f(x)$, the $x$-axis and the lines...
• 15N.1.hl.TZ0.12f: Find the area of the region enclosed by the graph of $y = f(x)$ and the $x$-axis for...
• 15N.1.hl.TZ0.12g: Show that...
• 15N.2.hl.TZ0.5a: (i)     Express the area of the region $R$ as an integral with respect to $y$. (ii)    ...
• 16M.1.hl.TZ2.11a: Calculate the value of the volume generated.
• 16N.2.hl.TZ0.10f: Consider the region $R$ enclosed by the graph of $y = f(x)$ and the axes. Find the volume of...
• 17M.2.hl.TZ1.4a: Write down a definite integral to represent the area of $A$.
• 17M.2.hl.TZ1.4b: Calculate the area of $A$.
• 17M.2.hl.TZ2.2b: Find the volume of the solid formed when the region bounded by the curve, the $x$-axis for...
• 17N.2.hl.TZ0.10d: This region is now rotated through $2\pi$ radians about the $x$-axis. Find the volume of...
• 18M.1.hl.TZ1.4a: $\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x}$.
• 18M.1.hl.TZ1.4b: $\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x}$.
• 18M.1.hl.TZ1.7b: Find $\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x}$.
• 18M.2.hl.TZ1.9d: The shaded region is rotated by 2$\pi$ about the $y$-axis. Find the volume of the solid formed.
• 18M.1.hl.TZ2.11c: The region R, is bounded by the graph of the function found in part (b), the x-axis, and...

Volumes of revolution about the $x$-axis or $y$-axis.

• 12M.2.hl.TZ1.8: A cone has height h and base radius r . Deduce the formula for the volume of this cone by...
• 08M.1.hl.TZ1.10: The region bounded by the curve $y = \frac{{\ln (x)}}{x}$ and the lines x = 1, x = e, y = 0 is...
• 08N.2.hl.TZ0.12: The function f is defined by...
• 09M.1.hl.TZ2.5: Consider the part of the curve $4{x^2} + {y^2} = 4$ shown in the diagram below.     (a)  ...
• 09N.1.hl.TZ0.10: A drinking glass is modelled by rotating the graph of $y = {{\text{e}}^x}$ about the y-axis,...
• 13M.2.hl.TZ1.4: Find the volume of the solid formed when the region bounded by the graph of $y = \sin (x - 1)$,...
• 10M.1.hl.TZ1.8: The region enclosed between the curves $y = \sqrt x {{\text{e}}^x}$ and...
• 10N.2.hl.TZ0.13: Let $f(x) = \frac{{a + b{{\text{e}}^x}}}{{a{{\text{e}}^x} + b}}$, where \(0 < b <...
• 11N.1.hl.TZ0.4c: find the volume of the solid formed when the graph of f is rotated through $2\pi$ radians...
• 11M.2.hl.TZ1.14a: When the glass contains water to a height $h$ cm, find the volume $V$ of water in terms of...
• 09M.2.hl.TZ1.12: (a)     If A, B and C have x-coordinates $a\frac{\pi }{2}$, $b\frac{\pi }{6}$ and...
• 14M.2.hl.TZ1.5b: The region $S$ is rotated by $2\pi$ about the $x$-axis to generate a solid. (i)     Write...
• 13N.1.hl.TZ0.12f: S is rotated through $2\pi$ radians about the x-axis. Find the value of the volume generated.
• 13N.2.hl.TZ0.13b: The domain of $f$ is now restricted to $x \geqslant 0$. (i)     Find an expression for...
• 13N.1.hl.TZ0.12e: Hence find the value of $\int_0^{\frac{\pi }{2}} {{{\cos }^6}\theta {\text{d}}\theta }$.
• 14N.2.hl.TZ0.13a: If the container is filled with water to a depth of $h\,{\text{cm}}$, show that the volume,...
• 15M.2.hl.TZ1.1: The region $R$ is enclosed by the graph of $y = {e^{ - {x^2}}}$, the $x$-axis and the lines...
• 15M.2.hl.TZ2.6b: The region bounded by the graph of $y = \ln (5x + 10)$, the $x$-axis and the lines...
• 15N.2.hl.TZ0.5b: Find the exact volume generated when the region $R$ is rotated through $2\pi$ radians about...
• 16M.1.hl.TZ2.11a: Calculate the value of the volume generated.
• 16M.1.hl.TZ1.13d: Find the volume of the solid formed when the region $R$ is rotated through $2\pi$ about the...
• 16N.1.hl.TZ0.11f: Find the area of the region enclosed by the graph of $f$ and the $x$-axis. The curvature at...
• 16N.2.hl.TZ0.10f: Consider the region $R$ enclosed by the graph of $y = f(x)$ and the axes. Find the volume of...
• 17M.1.hl.TZ1.11f: Determine the area of the region enclosed between the graph of...
• 17M.2.hl.TZ1.4a: Write down a definite integral to represent the area of $A$.
• 17M.2.hl.TZ1.4b: Calculate the area of $A$.
• 17M.2.hl.TZ2.2b: Find the volume of the solid formed when the region bounded by the curve, the $x$-axis for...
• 17N.2.hl.TZ0.10d: This region is now rotated through $2\pi$ radians about the $x$-axis. Find the volume of...
• 18M.1.hl.TZ1.4a: $\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x}$.
• 18M.1.hl.TZ1.4b: $\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x}$.
• 18M.1.hl.TZ1.7b: Find $\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x}$.
• 18M.2.hl.TZ1.9d: The shaded region is rotated by 2$\pi$ about the $y$-axis. Find the volume of the solid formed.
• 18M.1.hl.TZ2.11c: The region R, is bounded by the graph of the function found in part (b), the x-axis, and...