Area under a curve

The graph of y = e^x can be transformed to the graph of y = e^2x-1 by a stretch parallel to the x-axis followed by a translation.

State the scale factor of the stretch.

Give full details of the translation.

Alternatively the graph of y = e^x can be transformed to the graph of y = e^2x-1 by a stretch parallel to the x-axis and a stretch parallel to the y-axis.

State the scale factor of the stretch parallel to the y-axis.

The point P lies on the curve y = e^2x-1 and has x-coordinate of

Show that the tangent to the curve y = e^2x-1 at P has equation y = 2x.

Find the exact area enclosed by the curve y = e^2x-1, the tangent to the curve at P and the

y-axis.

Use the substitution u = cos x to show that

Fig. 1 shows part of the curve y = The shaded region is bounded by the curve, the x-axis, and the line x = a.

You are given that the area of the shaded region is a square units.

Show that the value of a satisfies the equation a - cos^-1(1 - 2e^-a) = 0.

Show by calculation that the value of a lies between 1.1 and 1.2.

Use the iterative formula

a_n+1 = cos^-1(1 - 2e^-a_n),

with starting value a₁ = 1.2, to find the value of a correct to 2 decimal places. Show the

result of each step of the iterative process.

Fig. 2 shows the curve y = cos^-1(1 - 2e^-x).

Explain why the gradient of the curve y = cos^-1(1 - 2e^-x) at the point where x = a, where a

is the value found in part (d), lies in the interval (- 1, 0).

The diagram shows the curve with equation y = (x³ - 2x²)ln x. The curve has a point of inflection at the point M.

Show that the x-coordinate of M satisfies the equation

Use an iterative formula, based on the equation in part (a)(i), to determine the x-coordinate of M correct to 2 decimal places. Use an initial value of 1.1 and show the

result of each step of the iterative process.

Determine the exact area of the shaded region, giving your answer in the form p ln q - r,

where p and r are positive rational numbers and q is a positive integer.

In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve y = , the two axes and the

line x = 2.5.

Amaya and Ben integrated (1 + x)², with respect to x, using different methods, as follows.

Charlie said that, because these answers are different, at least one of them must be wrong.

Explain whether you agree with Charlie’s statement.

You are given that a is a constant greater than 1.

In this question you must show detailed reasoning.

In this question you must show detailed reasoning.

Determine the area of the shaded region, giving your answer in the form p + q ln 2 where p and q

are constants to be determined.

The diagram shows the curve M with equation y = xe^-2x.

Show that M has a point of inflection at the point P where x = 1.

The line L passes through the origin O and the point P. The shaded region R is enclosed by the curve M and the line L.

Show that the area of R is given by

where a and b are integers to be determined.

The diagram shows the curve C with parametric equations

The maximum point on C is denoted by P.

Determine the exact coordinates of P.

The shaded region R is enclosed by the curve, the x-axis and the lines x = 1 and x = 6.

Show that the area of R is given by

where a and b are constants to be determined.

Hence determine the exact area of R.

The diagram shows part of the curve y = ln (x - 4).

State the equation of the vertical asymptote to the curve y = ln (x - 4).

Find the total area enclosed by the curve y = ln (x - 4), the x-axis and the lines x = 4.5 x = 7. Give your answer in the form a ln 3 + b ln 2 + c where a, b and c are constants to be

found.

Find the exact area of the shaded region, giving your answer in the form p + q ln r, where p and q

are positive integers and r is a positive rational number.

The diagram shows a part ABC of the curve y = 3 - 2x², together with its reflections in the lines y = x, y = - x and y = 0.

Find the area of the shaded region.