Math Problems 9/24/26

Question 1

Evaluate and graph the following definite integral.

\[\int_{0}^{3} \sqrt{9 - x^2} \space dx\]

# install.packages("tidyverse")
library(tidyverse)

## Warning: package 'lubridate' was built under R version 4.5.2

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f <- function(x) {
  sqrt(9 - x^2)
}
answer <- integrate(f = f,lower = 0,upper = 3)$value
x_values <- seq(0,3,length.out = 500)
y_values <- f(x_values)
q1_data <- data.frame(x = x_values,y = y_values)
ggplot(q1_data,aes(x = x,y = y)) +
  geom_line(col = "black",lwd = 1.25) +
  geom_ribbon(data = subset(q1_data,x >= 0 & x <= 3),
              aes(ymin = 0,ymax = y),
              fill = "blue") +
  labs(title = "Graph of f(x) = sqrt(9 - x^2)",
       caption = paste("Answer:",round(answer,4)),
       x = "x",
       y = "y") +
  theme_gray()

Question 2

Write 806 in words.

# install.packages("english")
library(english)

## Warning: package 'english' was built under R version 4.5.2

words(806)

## [1] "eight hundred six"

Question 3

Graph the corresponding polygon given the data below.

# install.packages("tidyverse")
library(tidyverse)
q3_data <- data.frame(X = c(-5,-5,4,4),
                      Y = c(-3,9,9,4))
ggplot(q3_data,aes(x = X,y = Y)) +
  geom_polygon(fill = "yellow",col = "black",lwd = 1.25) +
  geom_point(size = 4) +
  coord_equal() +
  theme_gray()

Question 4

Use integration by substitution to find \(\int \sin(x^2) \space 2x \space dx\).

# install.packages(c("ggformula","mosaicCalc"))
library(ggformula)

## Warning: package 'ggformula' was built under R version 4.5.2

## Loading required package: scales

## 
## Attaching package: 'scales'

## The following object is masked from 'package:english':
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## Loading required package: ggiraph

## Warning: package 'ggiraph' was built under R version 4.5.2

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## Warning: package 'ggridges' was built under R version 4.5.2

## 
## New to ggformula?  Try the tutorials: 
##  learnr::run_tutorial("introduction", package = "ggformula")
##  learnr::run_tutorial("refining", package = "ggformula")

library(mosaicCalc)

## Warning: package 'mosaicCalc' was built under R version 4.5.2

## Registered S3 method overwritten by 'mosaic':
##   method                           from   
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g <- makeFun(sin(x^2) * 2 * x ~ x)
anti_g <- antiD(g(x) ~ x)
anti_g

## function (x, C = 0) 
## {
##     F <- makeF(sin(x^2) * 2 * x)
##     evalFun(F, x = x, .const = C)
## }
## <environment: 0x0000022c9b26e690>