Definite integrals

Japanese school year: Math II

What you learn

This topic covers evaluating definite integrals by substituting the upper and lower limits into an antiderivative and taking their difference. It is widely applied in calculating geometric areas, volumes, and physical work. Prior familiarity with finding antiderivatives and algebraic substitution is necessary.

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Key points

This formula calculates the value of a definite integral from $a$ to $b$ . Find the antiderivative $F(x)$ of $f(x)$ , then subtract $F(a)$ from $F(b)$ .

\int_a^b f(x) \, dx = [F(x)]_a^b = F(b) - F(a)

This formula shows that integrals over adjacent intervals can be connected. Adding the integral from $a$ to $b$ and from $b$ to $c$ equals the single integral from $a$ to $c$ .

\int_a^b f(x) \, dx + \int_b^c f(x) \, dx = \int_a^c f(x) \, dx

This formula states that differentiating an integral from a constant $a$ to variable $x$ with respect to $x$ yields $f(x)$ . Use this when differentiating functions defined by integrals.

\frac{d}{dx} \int_a^x f(t) \, dt = f(x)

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Practice 1Coming soon