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...