Mathematics, 27.02.2021 23:50 danielle413

A curve is represented by the parametric equations x=t , y=3t^2 find the area under the curve from t=1 to t=2

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Answer from: azaz1819

$\displaystyle A = 7$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

BracketsParenthesisExponentsMultiplicationDivisionAdditionSubtractionLeft to Right

Pre-Calculus

Parametric to Rectangular Form Conversion

Calculus

Integrals - Area under the curve

Area of a Curve Formula: $\displaystyle A = \int\limits^b_a {f(x)} \, dx$

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Integration Rule [Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$

Step-by-step explanation:

*Note:

The area under the curve is essentially the definition of an integral.

Step 1: Define

Parametric

x = t

y = 3t²

1 ≤ t ≤ 2

Step 2: Rewrite

Rewrite parametric to rectangular form and change bounds of integration.

[Parametric] Substitute in t: y = 3x²[Parametric] Plug in values of t [Bounds]: 1 ≤ x ≤ 2

Step 3: Find Area

Integration.

Substitute in variables/function [Area]: $\displaystyle A = \int\limits^2_1 {3x^2} \, dx$ [Area] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle A = 3\int\limits^2_1 {x^2} \, dx$ [Area] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle A = 3(\frac{x^3}{3}) \bigg| \limit^2_1$ [Area] Evaluate [Integration Rule - Fundamental Theory of Calculus 1]: $\displaystyle A = 3(\frac{2^3}{3} - \frac{1^3}{3})$ [Area] (Parenthesis) [Fraction] Evaluate exponents: $\displaystyle A = 3(\frac{8}{3} - \frac{1}{3})$ [Area] (Parenthesis) Subtract: $\displaystyle A = 3(\frac{7}{3})$ [Area] Multiply: $\displaystyle A = 7$

Topic: AP Calculus AB/BC

Unit: Area under the curve

Book: College Calculus 10e

Answer from: Quest

It’s hard to understand what you put.. can you show a picture ?

Answer from: Quest

step-by-step explanation:

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