Definite integration and its applications：練習問題

Evaluate the definite integral $$\int_{0}^{1} (e^x - x^3) \, dx$$.

Find the area of the region bounded by the curve \(y = x^2 - 4x + 3\) and the x-axis.

Given that \(\int_{-1}^3 f(x) dx = 7\), find the value of \(\int_{-1}^3 (2f(x) - 3) dx\).

Find the area of the region bounded by the curve $$y = \frac{1}{x^2}$$, the x-axis, and the lines $$x = 1$$ and $$x = 2$$.

Evaluate the definite integral $$\int_{1}^{e} \frac{\ln x}{x} \, dx$$.

The rate of change of the temperature of a substance is given by \( \frac{dT}{dt} = 10 e^{-0.5t} \) degrees Celsius per minute. Find the total increase in temperature during the first 4 minutes.

Calculate the exact value of \( \int_{0}^{4} \frac{x}{\sqrt{1+2x}} dx \) using the substitution \( u = 1+2x \).

A curve is defined by \( y = x^3 - 3x^2 \). Find the area of the region bounded by the curve and the \(x\)-axis.

The curve $$C$$ is defined by the equation $$y = x^3 - 4x$$.

(a) Find the coordinates of the points where the curve $$C$$ intersects the x-axis.

(b) Find the coordinates of the turning points of the curve $$C$$. Hence, sketch the curve $$C$$, clearly indicating all intercepts and turning points.

(c) Calculate the total area of the region bounded by the curve $$C$$ and the x-axis.

Consider the function \( f(x) = \frac{x}{\sqrt{x^2+9}} \) for \( x \ge 0 \).

(a) Find the exact value of \( \int_{0}^{4} f(x) dx \).

(b) Let \( g(x) = f(x) + k \), where \( k \) is a constant. It is given that the area of the region bounded by the curve \( y = g(x) \), the x-axis, and the lines \( x = 0 \) and \( x = 4 \) is 10 square units. Find the value of \( k \).

(c) Using the substitution \( u = \sqrt{t^2+9} \), or otherwise, evaluate the definite integral \( \int_{0}^{4} \frac{t e^{\sqrt{t^2+9}}}{\sqrt{t^2+9}} dt \).