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

Answers

Answer 1

Answer:

[tex]\displaystyle A = 7[/tex]

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:                                                                                     [tex]\displaystyle A = \int\limits^b_a {f(x)} \, dx[/tex]

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

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

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

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

Topic: AP Calculus AB/BC

Unit: Area under the curve

Book: College Calculus 10e


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The solution of [tex]25x = 300[/tex] is 12.The given inequality is [tex]25x>300[/tex].

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Step-by-step explanation:

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Answer:

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Enter the answer in the box.

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Step-by-step explanation:

Median - The middle number in a set a data

First, you need to list the numbers least, on the left, to greatest, on the right.

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[tex]\Large \boxed{\sf 5}[/tex]

Step-by-step explanation:

Data set is 3, 3, 4, 5, 5, 5, 6, 8, 8

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if you get right answer you get good review on here

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ues that is the right answer
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