If ∫₀² ƒ(x) dx = π, ∫₀² 7g(x) dx = 7, and ∫₀¹ g(x) dx = 2, find the value of each of the following.
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d. ∫₀² √2ƒ(x) dx

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Identify the given integral and the constant multiplier: the integral to find is \(\int_0^2 \sqrt{2} f(x) \, dx\).

Recall the property of integrals that allows constants to be factored out: \(\int_a^b c \cdot h(x) \, dx = c \int_a^b h(x) \, dx\) where \(c\) is a constant.

Apply this property to the integral: \(\int_0^2 \sqrt{2} f(x) \, dx = \sqrt{2} \int_0^2 f(x) \, dx\).

Use the given value \(\int_0^2 f(x) \, dx = \pi\) and substitute it into the expression: \(\sqrt{2} \times \pi\).

Express the final answer as \(\sqrt{2} \pi\), which represents the value of the integral \(\int_0^2 \sqrt{2} f(x) \, dx\).

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Definite Integral and Linearity

The definite integral represents the net area under a curve between two points. It is linear, meaning constants can be factored out: ∫a^b c·f(x) dx = c·∫a^b f(x) dx. This property allows simplification when integrating scaled functions.

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Percorso guidato

05:43

Definition of the Definite Integral

Given Integral Values and Substitution

Known integral values can be directly used to find new integrals involving the same function. For example, if ∫₀² f(x) dx = π, then ∫₀² √2 f(x) dx = √2 · π by applying linearity, avoiding the need for re-integration.

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04:27

Substitution With an Extra Variable

Understanding Integral Limits

The limits of integration define the interval over which the function is integrated. Changing limits affects the integral's value, so it is important to note the interval when applying given integral values or combining integrals.

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