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Zorluk: KolayDefinite Integrals and Area Under Curves

What is the value of the definite integral ∫13(3x2−2x) dx\int_{1}^{3} (3x^2 - 2x) \, dx?

  1. A
    1212
  2. 1818Cevap
  3. C
    2020
  4. D
    2424

Cevap

18
Integrating 3x2−2x3x^2 - 2x gives the antiderivative F(x)=x3−x2F(x) = x^3 - x^2. Evaluating F(3)−F(1)F(3) - F(1) gives (27−9)−(1−1)=18−0=18(27 - 9) - (1 - 1) = 18 - 0 = 18.

Adım Adım Çözüm

1
Find the indefinite integral of the function
\int (3x^2 - 2x) \, dx = x^3 - x^2
Apply the power rule of integration to each term: \int 3x^2 dx = x^3 and \int 2x dx = x^2.
2
Evaluate the antiderivative at the upper limit x = 3
(3)^3 - (3)^2 = 27 - 9 = 18
Substitute the upper limit into the antiderivative.
3
Evaluate the antiderivative at the lower limit x = 1
(1)^3 - (1)^2 = 1 - 1 = 0
Substitute the lower limit into the antiderivative.
4
Subtract the lower limit evaluation from the upper limit evaluation
18 - 0 = 18
By the Fundamental Theorem of Calculus, \int_{a}^{b} f(x) dx = F(b) - F(a).

Anahtar Kavram

Fundamental Theorem of Calculus for Definite Polynomial Integrals
Tahmini Süre:1m 0s
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Konular

Definite Integrals and Area Under Curves

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