Edexcel A-Level Maths: Integration

The curve $C$C has equation $y = 6x - x^2$y=6x−x2 and the line $l$l has equation $y = 2x$y=2x.

(a) Find the coordinates of the two points where the line $l$l meets the curve $C$C. (3 marks)

(b) The line $l$l and the curve $C$C enclose a finite region $R$R. Use integration to find the exact area of $R$R. (6 marks)

(c) The curve $C$C also meets the $x$x-axis at the origin and at one other point. Find the exact area of the finite region enclosed between the curve $C$C and the $x$x-axis. (3 marks)

During a chemical reaction the mass of a product, $x$x grams, at time $t$t minutes is modelled by the differential equation

$\frac{dx}{dt} = \tfrac{1}{2}\,(10 - x), \qquad 0 \leqslant x < 10$dtdx​=21​(10−x),0⩽x<10

Initially there is none of the product present, so $x = 0$x=0 when $t = 0$t=0.

(a) Separate the variables and integrate both sides to obtain a relationship between $x$x and $t$t. (5 marks)

(b) Use the initial condition to show that $x = 10\left(1 - e^{-t/2}\right)$x=10(1−e−t/2). (3 marks)

(c) State, with a reason, the mass of product formed in the long term (as $t \to \infty$t→∞). (2 marks)

(a) Using integration by parts, find $\displaystyle\int x\,e^{2x}\,dx$∫xe2xdx. (4 marks)

(b) Hence evaluate $\displaystyle\int_0^1 x\,e^{2x}\,dx$∫01​xe2xdx, giving your answer in the form $\tfrac{1}{4}\big(a\,e^2 + b\big)$41​(ae2+b) where $a$a and $b$b are integers to be found. (4 marks)

Use the substitution $u = x^2 + 1$u=x2+1 to evaluate

$\int_0^1 \frac{x}{\left(x^2 + 1\right)^2}\,dx$∫01​(x2+1)2x​dx

giving your answer as an exact fraction. You must show the change of limits clearly. (6 marks)

The table below gives values of $y = \sqrt{4 + x^2}$y=4+x2​, correct to four decimal places, for five equally spaced values of $x$x.

$x$x	$0$0	$0.5$0.5	$1$1	$1.5$1.5	$2$2
$y$y	$2.0000$2.0000	$2.0616$2.0616	$2.2361$2.2361	$2.5000$2.5000	$2.8284$2.8284

(a) Use the trapezium rule with all five values to find an estimate for $\displaystyle\int_0^2 \sqrt{4 + x^2}\,dx$∫02​4+x2​dx, giving your answer to three decimal places. (3 marks)

(b) State, with a reason, whether the trapezium rule gives an over-estimate or an under-estimate of the true value of this integral. (2 marks)

A curve passes through the point $(1,\,3)$(1,3) and its gradient function is given by

$f'(x) = 6x^2 - 8x + \frac{4}{x^2}, \qquad x > 0$f′(x)=6x2−8x+x24​,x>0

Find $f(x)$f(x). (4 marks)