Integration (Edexcel International A Level (IAL) Maths: Pure 2): Exam Questions

Given

      ${\int }_k^5(2x-1)dx=20$

find the value of the positive constant $k$.

Evaluate

      ${\int }_1^5(4x+6x^2)dx$

The diagram below shows part of the graph of $y=4-(x-2{)}^2$.

Write down the values of $x$where $y=0$.

Show that       $4-(x-2{)}^2=4x-x^2$

Evaluate

      ${\int }_0^4(4x-x^2)dx$

Write down the area of the region labelled $R$.

The diagram below shows part of the graph of  $y=x^2-4x+3$.

Find the area of the shaded region labelled $R$.

Find the $x$-coordinates of the intercepts of the line with equation  $y=2$ and the curve with equation  $y=x^2-4x+5$.

Evaluate

      ${\int }_1^3(x^2-4x+5)dx$

The diagram below shows the graphs of $y=2$  and $y=x^2-4x+5$. Find the exact area of the shaded region $R$.

The diagram below shows the graphs of the line $y=6-x$  and  the curve $y=x^2$ .

Work out the $x$-coordinates of the points labelled $P,Q\mathrm{and}R$.

Work out the area of the shaded region.

The diagram below shows a sketch of the curves with equations

         $y=x^2-3x+4$      and   $y=4-x^2+2x$

Find the $x$-coordinates of the intersections of the two graphs.

Show that the area of the shaded region labelled $R$ is given by

         ${\int }_0^{\frac{5}{2}}(5x-2x^2)dx$

Use calculus to find the area of the shaded region labelled $R$.

Use the two diagrams below to show how rectangles can be used to give an upper and lower bound when estimating the area under a curve using the trapezium rule.

A student is estimating the area bounded by the curve $y=\mathrm{f}\left(x\right)$, the $x$-axis and the lines $x=a$ and $x=b$. 

The student intends to find the area of two rectangles of equal width in order to estimate the area as shown in the diagram below.

By drawing a sketch, show how the student’s estimate of the area can be improved while still using rectangles of equal width.

The diagram below shows the graph with equation $y=\sqrt{x}+4$.

The shaded area is to be estimated using the trapezium rule where $h=2$.

[i]

Write down the number of strips to be used.

[ii]

Write down the number of ordinates to be used.

[iii]

Complete the table of values for $y=\sqrt{x}+4$, giving values to three significant figures where appropriate.

Use the trapezium rule with all the values from the table above to find an estimate of the integral

         ${\int }_4^{16}(\sqrt{x}+4)dx$

giving your answer to three significant figures.

State, with a reason, whether your answer to part [b] is an overestimate or an underestimate.

The trapezium rule is to be used to estimate the integral

            ${\int }_{-2}^0{x}^3+8dx$

By completing the table of values below, use the trapezium rule to estimate the integral given above. 

i] By finding the exact value of the integral above, find the percentage error of  your estimate from part [a].

ii] Explain how you could increase the accuracy of your trapezium rule estimate.

Use calculus to find the value of

            ${\int }_4^9\frac{x^2+1}{\sqrt{x}}dx$

Given

      ${\int }_1^p$( $1+\frac{1}{x^2}$) $dx=\frac{15}{4}$

find the value of the constant $p$, where $p>0$.

The diagram below shows part of the graph of $y=2x+3x^2-2x^3$.

Find the area of the shaded region labelled $R$.

The diagram below shows part of the graph of $y=x\left(x-1\right)\left(x+2\right)$.

Find the total area of the two shaded regions.

The line with equation $5y=14-3x$ cuts the curve with equation  $5y=20-2x-x^2$  at the points $P$ and $Q$, as shown.

Find the $x$- and $y$-coordinates of the points $P$ and $Q$.

Find the exact area of the region labelled $R$, giving your answer in the form $\frac{a}{b}$, where $a$ and $b$ are integers to be found.

The diagram below shows the graphs of $y=x+2$ and $y=10x-x^2-16$.

Find the exact area of the shaded region.

The diagram below shows a sketch of the curves with equations

          $y=x^3+3$  and  $y=-x^3+2x+3$

Find the $x$-coordinates of the points of intersection of the two graphs.

Use calculus to find the total shaded area enclosed by the two graphs.

The trapezium rule is to be used to find an estimate for the integral

            ${\int }_4^8\mathrm{f}\left(x\right)dx$

The table below shows values for $x$ and $\mathrm{f}\left(x\right)$, rounded to three significant figures where appropriate. 

Using the values in the table find

[i]

an estimate for the integral using 2 strips,

[ii]

an estimate for the integral using 4 strips,

[iii]

an estimate for the integral using 8 strips.

Justify which of the estimates from part [a] will be the most accurate estimate for the integral.

The diagram below shows part of the graph with equation  $y=3+2x-x^2$

Use the trapezium rule with 5 strips to find an estimate for the shaded area, giving your answer to three significant figures.

Use the trapezium rule with $h=0.25$ to find an estimate for the area bounded by the curve with equation $y=1+2^x$, the $x$-axis and lines with equations $x=1\mathrm{and}x=2$. Give your answer to five significant figures.

The integral

         ${\int }_1^2(1+2^x)dx$

can be evaluated as 3.8854 to five significant figures. Using this as its exact value, calculate the percentage error of your estimate from part [a].

Use calculus to find the value of

            ${\int }_2^4\frac{x^3+\sqrt[3]{x}}{2\sqrt{x}}dx$

giving your answer correct to 3 significant figures.

Given

      ${\int }_q^{4q}5x\sqrt{x}dx=15066$

find the value of the constant $q$.

The diagram below shows part of the graph of  $y=4x-x^3$.

Find the area of the shaded region labelled $R$.

Without doing any additional calculation, explain why  ${\int }_{-2}^2(4x-x^3)dx$ must be equal to zero.

The diagram below shows part of the graph of  $y=x^3-2x^2-x+2$.

Find the total area of the two shaded regions.

The diagram below shows the graphs of  $4y=4x+17$  and  $y=3x^2-8x-16$.

Find the exact area of the shaded region.

The diagram below shows the graphs of  $y=16-2x$  and  $y=x^3-15x^2+66x-80$.

Find the total area of the shaded regions.

The diagram below shows the graphs of $y=12x-x^2-27$ and $y=x^2-14x+45$.

Find the area of the shaded region, $R$.

The diagram below shows a sketch of the graph of $y=x(x-6{)}^2$   $x\ge 0$

The graph has a local maximum point at $(2,32)$ as indicated on the diagram. 

Use the trapezium rule with 5 ordinate values to estimate the area shaded.

Using the appropriate working values from part [a], find an upper and lower bound for the area shaded.

Suggest a reason why using the trapezium rule in this case is not appropriate.

The diagram below shows the graph of

            $y=\frac{4-x^2}{\sqrt{x+2}},x>-2.$

Use the trapezium rule with six strips to find an estimate of the integral

            ${\int }_{-1}^1\left(\frac{4-x^2}{\sqrt{x+2}}\right)dx$

to five significant figures.

By using your calculator to find the exact value of the integral to five significant figures, find the percentage error of your estimate from part [a].