Integration：練習問題

Calculate the exact area of the region bounded by the curve \( y = 3x^2 \), the \( x \)-axis, and the lines \( x = 1 \) and \( x = 4 \).

A curve has the gradient function \( \frac{dy}{dx} = \frac{x+1}{\sqrt{x}} \). Given that the curve passes through the point \( (1, 5) \), find the equation of the curve.

Find the area of the region bounded by the curve \( y = x^2 - 4x \) and the \( x \)-axis between \( x = 0 \) and \( x = 2 \).
Note: In this interval, the curve lies entirely below the x-axis.

Find \( \int (3 \sin x - 2 \cos x) \, dx \).

Use the substitution \( u = x^2 + 1 \) to find the exact value of \( \int_{0}^{1} x(x^2 + 1)^3 \, dx \).

Use the substitution \( u = x^2 + 1 \) to find the integral \( \int 2x(x^2 + 1)^4 \, dx \).

Find the exact volume of the solid generated when the curve \( y = x^2 \) from \( x = 0 \) to \( x = 2 \) is rotated through \( 360^\circ \) about the \( x \)-axis.

Find the exact volume of the solid generated when the region bounded by the curve \( y = \frac{2}{\sqrt{3x+1}} \), the \( x \)-axis, and the lines \( x = 1 \) and \( x = 3 \) is rotated through \( 2\pi \) radians about the \( x \)-axis.

A curve has the equation \( y = 3x^2 - 4x + 2 \).
(a) Find the coordinates of the point on the curve where the gradient is 8.
(b) Calculate the area of the region bounded by the curve, the \( x \)-axis, and the lines \( x = 1 \) and \( x = 2 \).

The rate of change of the temperature, \( \theta \), of a liquid cooling in a room is modeled by the differential equation \( \frac{d\theta}{dt} = -k(\theta - 20) \), where \( k \) is a positive constant and \( t \) is time in minutes.
(a) Given that \( \theta = 80 \) when \( t = 0 \), express \( \theta \) in terms of \( k \) and \( t \).
(b) If the temperature of the liquid is \( 50^\circ \text{C} \) after 10 minutes, find the exact value of \( k \).
(c) Find the time taken for the liquid to reach \( 30^\circ \text{C} \), giving your answer to the nearest minute.